"Science, Faculty of"@en . "Mathematics, Department of"@en . "DSpace"@en . "UBCV"@en . "Parker, Sidney Thomas"@en . "2011-10-31T18:18:51Z"@en . "1934"@en . "Master of Arts - MA"@en . "University of British Columbia"@en . "[No abstract available]"@en . "https://circle.library.ubc.ca/rest/handle/2429/38465?expand=metadata"@en . "'OH 'D3V 'OH iVD BLAKE GOOEDIK&IES by Sidney Thomas Parker U . B . C J L J B R A R Y CAT. m.LElAlU^J&i&M ACC. NO! l?$JdL A Thesis submitted f o r the Degree of MASTER OF IRIS i n the Department THE IMIYERSITX OF BRITISH COLUMBIA. October, 1934 X \u00C2\u00BB TABLE OF CONTENTS CHAPTER I X \u00C2\u00AB Intr0duotion. 2 . Fundamental D e f i n i t i o n s . 3 . CoBrdinates of Planes. 4. P a r a l l e l P lanes. D i r e c t i o n Cosines of a L i n e . 6 \u00C2\u00AB Angle Between Two D i r e c t e d L i n e s . P o l a r Coordinates of a Plane. 8\u00C2\u00AB R o t a t i o n of Axes. ?. Standard Form of the Equation of a P o i n t . XO & Equations of P o i n t s (Continued). 11. Distance Between Two P o i n t s . 1 2 \u00C2\u00BB D i v i s i o n of a Segment i n a Given R a t i o . 1 3 . Plane Through Three P o i n t s . 14. The Expression^/ ( x x ~ x f )*\" + ' ( y v ~ Jt Y * 1 5 \u00C2\u00AB Distance Between P a r a l l e l \" P l a n e s . l 6 e Distance to a P o i n t from a Plane. If] e Angles Between Line and Planej Plane and PXane * 18 \u00C2\u00A9 Two-Point Equations of a Line\u00C2\u00BB 19* Equations of a Line (Continued). 20. Two-Plane Form of the Equations of a Line 210 D i r e c t i o n Cosines o f a Line . I I . 22. Plane P a r a l l e l to a L i n e . 23. P e n c i l of Planes. j 24. Three-Plane Equation of a P o i n t . 25.. T r a n s l a t i o n of Axes. 260, The Degree of an Equation i s Unchanged by Ro t a t i o n and T r a n s l a t i o n Transformations, CHAPTER I I - The general Second Degree Equation 1* Equation o f the Tangent P o i n t . 2. C o n d i t i o n that a P o i n t L i e s on the S u r f a c e 3 \u00C2\u00BB Locus of Middle P o i n t s of a System of P a r a l l e l Chords. 4. The P r i n c i p a l Plane* 5 . ' The Roots o f ( 1 7 ) . 6. E l i m i n a t i o n of the ys, az, z Terms. 7 . Reduction when d ^ 0 . 8. Reduction when d = 0. 9. Center of the Conicoido 10\u00C2\u00BB P o l a r Plane. 11. R e c t i l i n e a r Generators 12* Invariants\u00C2\u00A9 XXX \u00E2\u0080\u00A2 CHAPTER I I I C l a s s i f i c a t i o n of Surfaces 1. Review of Previous f o r i . 2. The Sphere. 3 . The E l l i p s o i d . 4. The Hyperboloid of One Sheet. J?. The Hyperboloid of Two Sheets. 6. The P a r a b o l o i d . 7 * I n v a r i a n t s f o r the Various Equations. CHAPTER IT Reduction of the General Equation 1. General Statement. 2. Reduction of the P o i n t - C o n d i t i o n Equation. 3 . To Find the Equation o f the Center of a Conic o i d . 4. The D i s c r i m i n a t i n g Cubic. 5 . D i s c u s s i o n f o r 0 . 6 . Discussion f o r J \u00C2\u00A9 *= 0 . 7 \u00E2\u0080\u00A2 Summary. BIBLIOGRAPHY PLAHE OOOBDIHAIES Chapter I 1. Introduction: -The primary purpose of t h i s t h e s i s i s to develop the ordinary r e l a t i o n s of s o l i d a n a l y t i c geometry \"by the use of plane-coordinates. The significance of var i o u s equations of the C a r t e s i a n system w i t h reference to t h i s new system w i l l also be d i s c u s s e d . As far as possible, the treatment parallels the treat-ment of line-coBr&lnates, as contained i n the theses sub-mitted, by ValgarcLsson of Manitoba and Heaslip and James of B r i t i s h Columbia for the degree of Master of Arts. 2 * Fundamental D e f i n i t i o n s t -We use the rectangular reference system, i.e. three mutually perpendicular planes intersecting in three mutually perpendicular straight lines x'ox, Y/0Y, z'oz, which are called the X, Y, Z axes,respectively. The X a x i s i s formed by the intersection of the ZX and XY planes; the Y axis by the intersection of the XY and YZ planes; and the Z axis by the intersection of the YZ and ZX planes. The point 0 , common to a l l three planes, i s called the origin. The cus-tomary conventions with regard to sign are observed. For example, the directions x' OX, Y'OY, Z'OZ are considered positive, and the directions XOX , YOY , 20Z are considered negative* The ooBrdinates of a plane are defined to he the re-ciprocals of i t s intercepts on the coordinate axes. Thus the plane ABO i n figure ( l ) has coordinates (a, b, c), since In Cartesian coSrdinates the point (a, b, e) is such that its. directed perpendicular distances from the YZ, ZX, XY planes are a, b, c, respectively. The plane ax * by * cz - 1 = 0 has intercepts i , - i . i on the coordinate axes* a D c 3\u00C2\u00BB CoOrdinates of P l a n e s ; -Any plane whose intercepts on the coordinate axes are a l l f i n i t e and different from zero i s seen to he repre-sented uniquely by (a, b, c). The following is a summary of some special cases: (i) Coordinate P l a n e s . The XT plane i s denoted by (a, b,<>0), where a and b are both f i n i t e . ( i i ) A plane through a coordinate axis and cutting the other axes obliquely. Such a plane through the X axis has the co8rdinates (a,oO where a i s f i n i t e . ( i i i ) The coordinates of a plane parallel to that given i n ( i i ) are (o, b, c), where b and e are f i n i t e . (iv) The ooOrdinates of a plane parallel to t h a t given i n ( i ) are (o, o, c), where o i s f i n i t e . (v) The \"plane at i n f i n i t y \" has the coordinates (o, o, o). (vi) A plane through the origin and oblique to a l l three axes has the coOrdinates (oO too>oo)\u00E2\u0080\u00A2 It is to be noted that the coordinates in ( i i ) and (vi) do not represent one plane uniquely, and that the planes i n ( i ) and ( i i ) do not possess unique coordinates. 4. Pa r a l l e l Planes:-Theorem: The necessary and sufficient conditions for the parallelism of two planes (x,, y, , z, ) and (x x, 7X \u00C2\u00BB z x) are (2) \u00C2\u00B11 = I I - l l . Fig. (2) The conditions are necessary. For suppose that the planes A, B, C, , i.e. (x ( , y( , z, ), and A^B^Cj, ,: i . e . U x , y^, z x ) , are p a r a l l e l . Then they cut the coordinate (1) planes in parallel lines, that i s , AjB, and A 2B t are pa r a l l e l . Hence OAV \u00E2\u0080\u009E 0Ba . OA/ OB/ (l) Wilson \"Solid Geometry and Conic Sections\u00E2\u0084\u00A2, p. 12. In the same way II * Z' ' Therefore X / B y ' OS Z ; x7 yT \"zl * The conditions are also sufficient. Suppose rela-tions (2) hold. Then A,B, is parall e l to A ^ , and B, G, is parallel to B^ O,.\u00E2\u0080\u00A2 Hence plane A,B,C, i s parallel to (1) plane A tB lC 3 L. This theorem i s equivalent to the statement that the planes (a, b, c) and (ka, kb, ke) are parallel* In the Cartesian system, two points whose coordinates satisfy equations (2) are eollinear with the origin, and conversely. If two planes: A,ac *\u00E2\u0080\u00A2 B, y * 0, z - 1 \u00C2\u00AB 0f A^x * B ^ * C^z - 1 = 0 are parallel, then A, - B, = 2\u00C2\u00B1 X~ B t U x ' and conversely. (1) Wilson, loc.oit., p. 13. j>\u00C2\u00BB Direction Gosines of a Mne:-'\"7 Let A be any directed line in space, and l e t be the line through the origin with the same direction as Let c< tfi , y be the angles between the X, Y, Z axes, res-pectively, and By definition these are the angles which makes with the axes. Ehey are called the \"direction angles\" of the line st- , and their cosines are called i t s \"direction cosines\". The direction cosines w i l l be denoted by A P /A, ^ respectively. z (1) As in Snyder and Sisam \"Analytic Geometry of Space\". ( 2 ) See Snyder and Sisam, p. 3 . It i s easily proved that the relation holds. CD 6* Angle between Two Directed Lines:' Suppose that and are two directed lines with direction cosines A, , AJ, , and A \u00C2\u00BBy) 7 oos r cos \u00E2\u0080\u00A2\u00E2\u0080\u00A2/* cos f. 8 \u00C2\u00AB Rotation of Axes;-12 Fig. (6) l e t the original reference system he rotated about the origin to a new position so that the new X axis has direction cosines A, t/U,, Vt , the new T axis has direction 12* eosines A,. ,yav > ^ \u00C2\u00BB and the new Z axis has direction cosines 3^ \u00C2\u00BB/S \u00C2\u00BB ^ \u00C2\u00BB a 1 1 wi\"ttL respect to the old axes. We shall denote the new axes by primed letters. Suppose the X' axis cuts any plane (x, y, z) at , as i n figure (6). Denote the angle POAx by 0 . By equa-tions (4), the direction cosines of OP are cos o< = cos cos Y From equation ( 3 ) we obtain. ._ \u00E2\u0080\u00A2 X . .. \u00E2\u0080\u00A2i/.x1^. * y y * z1\" V x * y \u00C2\u00AB \u00E2\u0080\u00A2. , .. ,..,Z : . \u00E2\u0080\u00A2 z v x ^ * y 1 in * A, x +^.y y x + y But, from figure (6), i t follows that A OP x COS. U st \u00E2\u0080\u00A2QJ\"*/ SB - / x 1 \" * y 2 ^ z i By equating these two values for cos 0 , we get similarly (6) , y = A x * + / \ / * <2 , and 1 3 . The inverse transformations are 17) y = yA\u00C2\u00AB *' \u00E2\u0080\u00A2* + /u) x'} Y/e can express results (6) and (7) i n tabulated form as follows; ( 8 ) 2/' J 2 These relations are exactly the same as those ob-tained for 0artesian coOrdinates* 9. Standard fform of the Equation of a Point;-The standard equation of a point w i l l be that rela-tion which involves the directed perpendioular distances from the three coordinate planes to the point. Let P be the point whose directed perpendicular distances from the YZ, ZX, and XY planes are r, s, and t respectively. In figure l 7 ) s OR \u00C2\u00AB r, EQ, = s, QJ? = t. Rotate the axes so that the x' axis passea through P. Then i s the x' coOrdinate of a l l planes which pass through P. Therefore Fig. (7) But, from (6), we have where Ax \u00C2\u00AB\u00E2\u0080\u00A2 0 R = 19) Therefore s 4/1 r * s * t y r * s + t and hence U P ) rx * sy \u00C2\u00BB tz r + s r + s + rx +\u00E2\u0080\u00A2 sy * tz - 1 = 0 , 15. We must now show that a l l planes whose coordinates satisfy (10) pass through the given point. Let (a, b, c) \"be a plane whioh does not pass through P, hut whose coordinates satisfy (10)\u00E2\u0080\u00A2 Then (11) ra * sb .* tc - 1 * 0. From section 4, the coordinates of a plane through P and parallel to (a, b, c) are (lea, kb, ko). Since these coordi-nates must satisfy ClO), i t follows that (12) k(ra * sb * tc) 1 \u00C2\u00AB 0. The equations ( l l ) and (12) are both true only i f k = 1, i n which case the plane (ka, kb, kc) i s coincident with the plane (a, b, c). Therefore the plane (a, b, c) must pass through the point. 10. Equations of Points (Continued):-The standard equation of a point P i s given by (10)\u00E2\u0080\u00A2 The direction cosines of OP are given in (9). If we denote the length of OP by f ,. equation (10) may be written (13) A x + yu.y --L a 0. We shall c a l l (13) the \"directed\" equation of the point. If (^ , e< t / s , y) are the polar coordinates of a plane, whose intercept coordinates are (x, y, z), passing through the point rx * sy * tz - 1 \u00C2\u00BB 0 then 16. JL5 . + \u00C2\u00A7y . tz + y v * z\" f/x\" * y\"\" + /x\"- * y\"~ * z1\" V ? \u00E2\u0080\u00A2*\u00E2\u0080\u00A2 y*\" * z' Therefore (14) ^ = r oosc< * a cos/? + t cos \u00C2\u00A5 We shall e a l l (14) the \"polar I r equation of the point. The equation of the origin i s ox * oy + os - 1 \u00C2\u00BB 0, The equation of the \"point at i n f i n i t y \" i s Ax \u00E2\u0080\u00A2-t-yu-y -t- V-z-The equation of a point on the X axis is : TX - 1 s o, and the equation of a point i n the XT plane i s rx t sy \u00E2\u0080\u00A2 1 B 0, In Cartesian coordinates the plane ox * oy * oz - 1 = G ClV i s known as the \"plane at i n f i n i t y \" . ' The plane Ax + yu. y +1/2. tt 0 passes through the origin and ^ 9^u,t 4/are the direction cosines of the normal to the plane. The plane rx - 1 = 0 is parallel to the YZ plane* ( l ) Snyder and Sisam, p. 34* 17. 11* Distance \"between Two Points:-and Let two points P; and P x be denoted by the equations r f s * st y * t,z - 1 = 0 Fig. (8) Let the lengths of P, \ , OP, , OP^ be d\u00C2\u00BB 9 fK res-pectively, and let angle P; OP^be Q \u00E2\u0080\u00A2 We have and hence, from ( 3 ) and (?), i t follows that d\"\" * ( r # v * a,* * t,\"*\" )+(r ~ * sj\" * t ^ ) - 2 ( r , r f c + s ( s z +t,t a), so that (15) a =y ( r t - r ) % (a,. ^ S / ) % ( t ^ - t r. 1 8 . 12\u00E2\u0080\u00A2 Division of a Segment in a Given Ratio:-Let the segment be defined by the two points given in section 11, and let the given ratio of division be h : lu Suppose that P, the division point, has the equation rx * sy + tz - 1 = 0\u00C2\u00AB Hence On solving for r we obtain 1 9 . ( 1 6 ) Similarly t = _ 3cs, * hs. - \u00E2\u0080\u00A2 h + 1c let, + ht 3 1 3 . Plane through. Three Points Let the equations of the three distinct points P; , P 2, ? 3 be r, x * s,y * t,z - 1 \u00C2\u00AB 0 , r z x * szj * t^z - 1 = 0 , r ^ i \u00E2\u0080\u00A2 ajy.* t 3 z f 1 \u00E2\u0080\u00A2 0 , respectively. If these equations are solved for x, y, z, we obtain the coordinates of a plane passing \"through the three points. Finite solutions are possible provided that r, s. t, s, t % r3 s, t, If .A ~ 0, then each element of any one row i s a linear com-bination of the corresponding elements of the other two rows. Suppose that r 3 \u00C2\u00AB= k,r, + t 3 = x,t, + k.t,. Let us consider the point P whose equation is \u00E2\u0080\u00A2rx * ay + tz > 1 \u00C2\u00BB 0, where 2 0 . r = ~ l + 3c xr x + + From (16) we see that the point P is collinear with P, and P r\u00C2\u00BB Therefore any plane through P, and P z must pass through P. \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 From (9) we see that the vectors OP and 0P3 are one and the same straight l i n e . Therefore the origin, P, and Pj are collinear. Hence, any plane passing through P and P3 must pass through the origin, and one, at least, of x, y, z must be i n f i n i t e . In Cartesian coordinates three planes determine a point except when one plane is parallel to the line of inter-section of the other two. The condition for this exception is /\ .** 0. 14. The Expression 1/(x, - x, ) x + (y t - y, )*\" + ( z t - z( Let \u00C2\u00A9 be the angle between the perpendiculars from the origin to two planes (xf , y, , z, ) and (x a, y^ , z 2) and let d be the distance between the feet of these perpendi-culars. Then d1\" \u00C2\u00AB* \u00E2\u0080\u00A2 f>* + - 2f, / \ QOS Q , where -ft and A are the lengths of the polar normals as given i n (4) and cos Q i s determined by the r e l a t i o n ( 1 7 ) cos 0 \u00C2\u00BB\u00E2\u0080\u00A2 .\u00E2\u0080\u00A2j/ x, v + y,v + 2\"\u00E2\u0080\u00A2 / x j \" + yj\" + z^ that i s . -j i ( x x + y y + z z ) x , + y, + z, -x. + y- + 0*. + y, + z, + y t +zj or ,x x,\" + y,\" -J- z,v + x^ + yj- \u00E2\u0080\u00A2 -2(x, x^ + y, y^ + z, z j \u00E2\u0080\u0094 \u00E2\u0080\u0094 \u00E2\u0080\u0094 \u00E2\u0080\u0094 \u00E2\u0080\u0094 \u00E2\u0080\u0094 \u00E2\u0080\u0094 \u00E2\u0080\u0094 \u00E2\u0080\u0094 \u00E2\u0080\u0094 \u00E2\u0080\u0094 \u00E2\u0080\u0094 _ \u00E2\u0080\u0094 \u00E2\u0080\u0094 \u00E2\u0080\u0094 \u00E2\u0080\u0094 \u00E2\u0080\u0094 \u00E2\u0080\u0094 \u00E2\u0080\u0094 ^ + y,\" + z,1- ) ( x V * yj- + ) which reduces to (IS) d. - - x ) % (y x - y, ) X + (z\u00E2\u0080\u009E -z, f . 1 3 . Distance between P a r a l l e l P l anes:-The distance between the p a r a l l e l planes (x, y, z) and (lex, Icy, kz) i s equal to the dis t a n c e between the f e e t of t h e i r p o l a r normals. From equation ( 1 8 ) we ob t a i n ( 1 9 ) 3c - X x + y + z 1 6 . Distance to a P o i n t from a Plane:-Let the poin t be defined by the equation r x + sy + t z - 1 = 0 and the plane by the coordinates (x ( , y( , z , ) . Through the p o i n t draw a plane w i t h coordinates, say, (toe, , Icy; , k.z( ), p a r a l l e l to the given plane. Then the distance to the 22 p o i n t from the plane i s equal t o the distance between these two planes. Since the new plane passes through the gi v e n p o i n t , we have k ( r x , + sy, + t z , ) - l = 0 ; that i s k - I r x , + sy, + t z On s u b s t i t u t i n g t h i s value f o r k i n (19) , we o b t a i n ( 2 0 ) d rx, + sy, + t z , - 1 Theorem: Two p o i n t s P, , P t whose equations are r, x + s, y + t, z - 1 = 0 , i \ x + s,y + t t z - 1 = 0 , r e s p e c t i v e l y , are on the same side or on opposite sides of the plane (x, , y , z, ), acc o r d i n g as i t s coordinates g i v e the f i r s t members of the equations of the p o i n t s l i k e or u n l i k e s i g n s . For, l e t the p o i n t of i n t e r s e c t i o n of the l i n e P, P^ and the plane be P whose equation i s r x + sy + t z - 1 = 0 , where r \u00C2\u00AB= m, r, * m^r^ ^ s = m.s, i m ta 4 , t = m,t, + m t t x } and m( + m^ = 1 ( S e c t i o n 1 2 ) t 23. Therefore (m, r, + m r )x + (m s + m a )y + (m t + m t )s - 1 = 0 ? that i s , m( ( r x + s, y, + t, z - 1 ) + m^(ru x, + s^y, + t i Z < - l ) = 0 . I f r, x, + s, y, + t, z( - 1 and r v x , + s^y, + t ^ z , - 1 have u n l i k e s i g n s , then m, and mx have the same s i g n , and the p o i n t P l i e s between P ( and P z . I f r, x,+ s, y, + t, z, - 1 and r ^ x, + s vy ( + t l z / - 1 have the same s i g n , then the numbers m,, mt have opposite s i g n s , hence the p o i n t P i s not between P, and P x . A p o i n t whose equation i s r x + sy + t z - 1 \u00C2\u00BB 0 w i l l be considered to be on the p o s i t i v e or negative side of the plane (x,, y , zt ) according as the expression r x , + sy, + t z , - 1 i s p o s i t i v e or negative r e s p e c t i v e l y . From (20) and the theorem Just proved we can say that the distance to a p o i n t from a plane i s p o s i t i v e or negative according as the p o i n t and the o r i g i n are on the same side or on opposite s i d e s of the plane. 1 7 . Angles between Line and Plane; Plane and Plane:-The angle between a l i n e and a plane i s the comple-ment of the angle between the l i n e and the p o l a r normal to the plane. I f A,yu, 1/ are the d i r e c t i o n cosines of a l i n e which makes an angle Q w i t h the plane (x, y, z ) , then from 24. (3) a M (4) we get (21) s i n 6)= / U \u00E2\u0080\u00A2 y U y + V z ^ / x ^ + y \" + z x The angle between two planes i s equal 10 the angle between t h e i r p o l a r normals and i s g i v e n by ( 1 7 ) . 18. Two-Point Equations of a L i n e : -Two d i s t i n c t p o i n t s w i l l determine a s t r a i g h t l i n e s i nce the t o t a l i t y of planes, which pass through the two p o i n t s s i m u l t a n e o u s l y , d e f i n e a l i n e . Hence the simultaneous equations (22) r,x + 3,7 * t, s - 1 \u00C2\u00AB 0 , r x x + s 2 y + t z z - 1 = 0 , give the equations\" of the l i n e . , We s h a l l r e f e r to (22) as the \"Two-Point\" equations of a l i n e \u00E2\u0080\u009E 1 9 . Equations of L i n e s (Continued);-The most general equations of a l i n e are g i v e n by ( 1 ) (22)\u00C2\u00BB The f o l l o w i n g i s a summary o f s p e c i a l esses: ( i ) A coordinate a x i s . The X a x i s has the equations r x - 1 = 0; ox + oy + oz - 1 = 0 . (1) I t i s understood t h a t r , s, and t are not zero i n the f o l l o w i n g work. 25\u00C2\u00BB ( i i ) A l i n e p a r a l l e l to ( i ) and passing through the Y a x i s has the equations x = 0 , s y - 1 \u00C2\u00AB= 0 . ( i i i ) A l i n e p a r a l l e l to ( i ) and p u t t i n g the YZ plane has the equations x = 0 , s y + t z - 1 \u00C2\u00AB= 0 . ( i v ) A l i n e \" t h r o u g h the o r i g i n and l y i n g i n a coordinate plane. Such a l i n e i n the XY plane has the equa-t i o n s r x + s y - 1 = 0 , ox+oy+oz-1 <= 0 . (v) A l i n e \" through the o r i g i n oblique to a l l three axes has the equations rx+sy+tz - 1 = 0 , ox+oy+oz-1 = 0 . ( v i ) A l i n e through the X and Y axes but not through the o r i g i n has the equations r x - 1 = 0 8 s y - 1 = 0 . ( v i i ) A l i n e through the X a x i s and p a r a l l e l to the YZ plane has the equations r x - 1 = 0 , rx+sy+tz - 1 \u00C2\u00AB=\u00E2\u0080\u00A2 0 . 26 20. Two-Plane Form of the \u00E2\u0080\u00A2Equations of a L i n e ; -Let the l i n e be defined by the planes (x,, y,, zf) and (x , y , z )\u00C2\u00BB I f the l i n e passes through the o r i g i n then one or more of the coordinates of each plane w i l l be i n f i n i t e . I f i t does not pass through the o r i g i n , a l l the members of at l e a s t one set of coordinates w i l l be f i n i t e . Suppose the p o i n t s r ( x + s ( y + t ( z - 1 = 0, r^x + s x y + t a z - 1 \u00C2\u00AB 0 , l i e on the l i n e . The p o i n t i n which the l i n e cuts the XY plane can be found by e l i m i n a t i n g z from the two equations, and the p o i n t where I t cuts the TZ plane can be found by e l i m i n a t i n g x. Let these two p o i n t s be denoted by the equa-t i ons (23) Then r^x + s 3 y - 1 = 0, s^y + t^z - 1 \u00C2\u00AB= 0V r e s p e c t i v e l y r 3 x + 3sy - 1 = 0, r,x, + s 3y, - 1 - OV r 3 x % * s 3 yx - 1 0. I f these equations i n r , s are to be c o n s i s t e n t we must have x y 1 x, y, 1 x x 5V 1 27. whence x - x, c y - y, - 2 r v y, - y v In the same way, from the second of\" equations (23) we o b t a i n y - y, = z - z, Therefore (24) x - x, ^ y - y ( & z ~ z, x, - x x y, - y x. z, -Equations (24) are c a l l e d the \"Two-Plane f r equations of a s t r a i g h t l i n e . Obviously these have no meaning i f one of the denominators i s zero. Suppose x, - x,. Is zero. Then x must be equal t o x i and i n s t e a d of (24) we w r i t e x - x ( , . . 7 - J , = z - z, y, - y x I n C a r t e s i a n coordinates (24) give the \"two-point\" equations of a s t r a i g h t l i n e . 21. D i r e c t i o n Cosines of a l i n e : -I f the lin e - i s define d by the two p o i n t s whose equations are (22), the d i r e c t i o n cosines are found t o be Suppose the l i n e i s defined by (24). Equate the f i r s t two f r a c t i o n s . Then (x - x,) (y, - y j - (y - y ( ) (x, - x j . This equation i s r e d u c i b l e to the form ( 2 6 ) y * \" y>_x + X/ \" y - 1 - 0, x , y v - x^y ;- x y v -x^y # which i s the equation of a p o i n t on the line\u00C2\u00BB In the same way the equa t i o n s ( 2 7 ) Z ^ \" z > y * y> ~ z - 1 = 0 , y, Z , ~ K Z , - y, 2 ^ z , and ( 2 8 ) g v \" z\u00C2\u00AB x X ' ~ ^i- E ~ 1 ~ o; X, Z -X Z , X z, -x z represent p o i n t s on the l i n e . We can therefore s e l e c t two of these p o i n t s and f i n d the d i r e c t i o n cosines of the l i n e j o i n i n g them by means o f ( 2 5 ) \u00E2\u0080\u00A2 I f the denominator x, y^ - x^y f has the value zero, i . e 0 , i f B y, ' y r 8 \u00C2\u00A3 9 from s e c t i o n 3 we know that x # z v - K , _ z ( 7, 2^ - y x z f cannot also be zero. In t h i s case we can use the two p o i n t s whose equations are ( 2 7 ) and ( 2 8 ) . 2 2 . Plane P a r a l l e l to a l i n e : -Theorem: The plane ( 2 9 ) (k,x, + k^^, k y, k v y ^ , k z, + k v z J i s p a r a l l e l t o the l i n e determined by the planes ( x / 9 y ( , z ( ) and ( x t , y L , z J . I f 6 I s the angle between the l i n e and p l a n e 9 from equation ( 2 1 ) we o b t a i n s i : y 1\" -5- y + z Let ( 2 7 ) and ( 2 8 ) be the equations of the l i n e . Then, from ( 2 5 ) we have 2 , \" Z , \u00E2\u0080\u00A2rr FT \u00C2\u00AB*. -v* \u00C2\u00BB7 // z v - z, y / z , - 2 , - ^ y, - y* 11-1/ (x, z v - x v s , j >, z v - y tz,/ \"fx, 7, 5L. z, / z / - z. (30) y, z t ~ y x z , z v \" s , ,v /z, ~ s v f /x, - x u y. ~ y\u00E2\u0080\u009E Ax z - X x z f c- x j a , / (y, z^- y^z,/ x_z ( y # z^- y^z 30 *v. y, - y. / x z - x z y z - y z ix, Z l- x x z ^ Z l\u00C2\u00BB y x z / U , V y,s\u00E2\u0080\u009E- y ^ , The s u b s t i t u t i o n of (30) i n the expression f o r s i n 0 gives us P + Q, + R s i n 6 - s.T . where -n E, \u00E2\u0080\u0094 Zi x , z r - x v z (k,x, + Ic^xJ Q . - J = (x,y, - \ y t , S - y (3c, x, + k,x v)\" + (x, y, + \ y ^ ) v + (x, z + IcjOV T A x z v - x^z ( / \y, \~ y^J \2c,z^- x k z ; y ( y z X. f The numerator reduces to zero and hence s i n g> \u00C2\u00AB 0, and the plane i s p a r a l l e l to the l i n e . Conversely, i f the plane ( x 3 , y^ , z^) i s p a r a l l e l to the l i n e of i n t e r s e c t i o n of (x f , y ( , z,) and ( x z , y L , z t ) , i t s coordinates must be o f the form (29)\u00E2\u0080\u00A2 We have 3 1 . and. t h e r e f o r e z - z, a: z - x z z - z t T-Xj + y 3 'x, - x ^ y, - y\u00E2\u0080\u009E X z - X z / V 2-1 y, Z T - ~ y * I s ^ - 0 , T l i i s eauation reduces to ( 3 1 ) x, y \u00E2\u0080\u00A2 i 2-x 3 y> z. 0 e (^z^ - z () cannot always be zero, since we do not have to r e s t r i c t the l i n e i n t h i s manner. (Therefore we must have the r e l a t i o n I aci- y', z,\ ( 3 2 ) x , y^ z. * i y, s3 = 0 s a t i s f i e d under a l l c o n d i t i o n s . I f ( 3 2 ) h o l d s , then x^, y 3 \u00E2\u0080\u009E z j must be a l i n e a r combination of the corresponding e l e -ments of the other two rows, and hence must be of the form (29).. In C a r t e s i a n coordinates a p o i n t ( 2 9 ) i s co-planar w i t h the p o i n t s (x, , y, , z( ), y t, z J and the o r i g i n . 23. P e n c i l of P l a n e s : -Suppose the plane ( 2 9 ) passes through the l i n e of 32* i n t e r s e c t i o n of the planes (x f, y, , z ( ) and ( x ^ y^, z u ) \u00E2\u0080\u00A2 Then i t passes through a l l p o i n t s on the l i n e and i t s c o o r d i -nates must s a t i s f y the equation of any poin t on the line\u00C2\u00AB, Let a poin t on the l i n e be defined by the equation rx + sy + t z - 1 = 0. We 'must have \u00E2\u0080\u00A2 r x , + sy, + t s , - 1 = 0, (33) rx^* sy1_+ t z t - 1 \u00E2\u0080\u00A2\u00C2\u00BB 0, r ( l c , x , * x^x u) + s(k,y,+ k j j + t(]c z ( + \ z j - 1 = 0, t h a t i s ' - ' (34) lc, ( r x ( * sy, + t z , ) + k r(rx v+ sy^_+ t z j - 1 = 0. Equations (33) and (34) h o l d simultaneously only i f (35) ^ = 1. This r e l a t i o n i s the necessary and s u f f i c i e n t c o n d i t i o n t h a t a plane,. whose coBrdinates are gi v e n by (29), w i l l pass through the l i n e of i n t e r s e c t i o n of the planes (x,, y , z, ) and ( x ^ , y^, z j . In ( 2 9 ) , i f we l e t x, = \u00E2\u0080\u0094\u00E2\u0080\u0094 s h + x h + 1c we have the system of planes whose coordinates are given, by 3cx + hx x \u00C2\u00AB* _ ! , h + x 33. (36) = Icy, * h y w . Ii * 3c 3cz . + h z . h + 3c which i s a p e n c i l o f planes, s i n c e r e l a t i o n (33) s t i l l h o lds. In C a r t e s i a n coordinates a l l p o i n t s (56) are c o l l i -near, and d i v i d e the segment j o i n i n g (x( , y ( , zt ) and (x^, y^, z^) i n the r a t i o h : 3c. 24. Three-Plane Equation of a P o i n t ; -\u00C2\u00A3et (x( , y( , z( ), (x^, y^, z.J, and ( x 3 , y f, 2 j) he the coordinates of three planes such t h a t no plane i s p a r a l l e l to the l i n e of i n t e r s e c t i o n of the other two\u00C2\u00AE The c o n d i t i o n s that these three planes pass through the p o i n t \ rx + sy + t z - l = 0, are rx, + sy, + tz, - 1 \u00E2\u0080\u00A2\u00C2\u00BB 0, rx^+ sy t + tz,.- 1 \u00C2\u00AB 0, rx 3 + sy, + tz,- 1 = 0. The c o n d i t i o n t h a t r , s, t e x i s t so as t o s a t i s f y these f o u r simultaneous equations i s that x y z 1 (37) z , 1 1 1 = 0. 3 4 This i s the r e q u i r e d equati o n t since i t i s of the f i r s t degree i n x, y, z, and i s obviously s a t i s f i e d by the coordinates of the three planes. I f ^ \u00C2\u00B0i \u00E2\u0080\u00A2.. . f] the p o i n t i s f i n i t e . I f w - 0 , ( 3 7 ) gives an equation of the form r.i r x + sy + t z = 0 , which has a l r e a d y been d e f i n e d as a p o i n t a t i n f i n i t y . I f CO \u00C2\u00AB= 0 , the elements of any one row o f OJ must be a l i n e a r combination of the corresponding elements of the other two; rows, and hence the plane must be p a r a l l e l to the l i n e of |J i n t e r s e c t i o n of the other two* hi \u00E2\u0080\u00A2N \u00E2\u0080\u00A2, '' H 25 \u00E2\u0080\u00A2 T r a n s l a t i o n of Axes;- . kj Suppose the 0 r i g i n i s t r a n s l a t e d t o the point ; ,! r x + sy + t z - 1 Oy without any r o t a t i o n of axes. Let any plane be represented by the p o l a r coordinates (f>} otf fi} y ) and ( ^ ' *, f' ) w i t h respect t o ihe o r i g i n a l and new systems, r e s p e c t i v e l y . Then ii; 2c, y, z, x ^ ^ x i y 3 z a 3 5 \u00C2\u00AB From (20) we have - (rx sy + tz - l ) f x + y + z Therefore 0 8 ) and hence (39) 1 x y' a' (40) ~ \u00E2\u0080\u00A2\u00C2\u00BB ( r x + sy + tz - -- A \u00C2\u00AB= y. ,* X r x + sy + tz - 1 - - y\" r x + sy + tz - 1 z r x + sy + tz - 1 ons are X rx' + sy' + tz' t 1 y' r x ' + sy' + tz' + 1 z' rx' + sy* + tz' + 1 260 The Degree of an Equation- i s Unchanged by Transf orma-t i o n s :- ^ ( 1 ) Tanner- and A l l e n \" A n a l y t i c Geometry\", p. 1 2 7 \u00E2\u0080\u00A2 Wentworth \" A n a l y t i c Geometry\", p. 109 \u00C2\u00BB 3 6 Let the degree of the equation he n. A general term would be (41) i x W , where p, q, m are not negative and p \u00E2\u0080\u00A2*- q + m \u00C2\u00A3 n. I f we r o t a t e axes by equations ( 7 ) , i n place of (41) we o b t a i n A(/),x'*Ay' + / U') P(/<.X'* /-.Y+yO, -0*(<*Wy'+ S 3 Z ' T Since each term i n each bracket i s o f the f i r s t degree, we cannot o b t a i n terms of degree higher than n. I f we t r a n s l a t e axes a c c o r d i n g to equations (40), (41) becomes A x ' P y'* z' m ' ( r x ' * sy* + t z ' ) P + <1 + m I f every term i n the new equation be m u l t i p l i e d by (rx' + sy' + t z ' + l ) n , the term (41) f i n a l l y becomes (42) A x ' P y'* z ' m ( r x ' + sy' + t z ' + l ) n ~ ( p + * * m ) Any term i n (42) cannot be o f degree higher than n. Hence the degree of an equation i s not r a i s e d by t r a n s l a t i o n or r o t a t i o n of axes* Suppose the degree were lowered by a transformation of c o o r d i n a t e s . Then, by applying the i n v e r s e transformation, 3 ? . we should, be r a i s i n g the degree o f the equation. This has been proved impossible. Therefore the degree i s unchanged by r o t a t i o n and t r a n s l a t i o n . 3 8 . CHAPTER I I The General Second Degree Equation The most gen e r a l second degree equation i n x, y, z i s ( l ) a x % hy w+ cz u+ 2fyz + 2gzx + 2hxz + 2ux + 2vy \u00E2\u0080\u00A2i- 2ws + d!, \u00C2\u00AB 0, \" where a t l e a s t one of a, b, c, f , g, h i s d i f f e r e n t from zero. We s h a l l show t h a t ( l ) always represents a c o n i c o i d i n the p l a n a r system of coordinates. 1. Equation of the Tangent P o i n t The l i n e of i n t e r s e c t i o n of the planes (x,, y , z, ) and ( x t , y^, zj) i s g i v e n ( S e c t i o n 20, Chap. I) by the equa-t i o n s .(2) x - x , = y - 7 | ^ z - 2 , = ^ - x,_ y, - 5^ Z / - z v The coordinates of any plane through (2) are x \u00C2\u00AB. x, + p(x, - x x ) , (3) . y = y, + p(y, - yj\u00C2\u00BB z = z, + p(z, - z J . I f a plane ( 3 ) touches the surface ( l ) , i t s coBrdinates must s a t i s f y equation ( l ) . S u b s t i t u t i n g ( 3 ) i n ( l ) we o b t a i n a quadratic equation i n p, which shows t h a t , i n general, through any l i n e two planes can be drawn to touch the surface ( 1 ) . Suppose that one of these i s the plane (x,, y(i z ) I t f o l l o w s that one r o o t o f the quadratic i n p must be zero, and hence the constant term must be zero. We ther e f o r e have (4) a x ( % by,\"* c z , \" * 2fy,z,+ 2gs( x, + 2hxty( + 2ux(+ 2vy, * 2w z , + d = 0\u00C2\u00AB Suppose (3) determines one plane only. In t h i s cas the plane i s the tangent plane (xt , y ( , zt ), and both roots of the quadratic are zero. Both the constant term and the c o e f f i c i e n t of p must be zero, so that ax, ( x , ~ x j + by, (y, - y j + cz, (z,- z j + f{ y, (z,- zj (3) * z, (y, - y j j * g f z( (x, - x j + x, (z, - z^ ) J + h{ x t ( y # - y^ ) + y, (* - ) / + u(x,~ x j + v(y, - yO + w(z, - z j = 0. I t f o l l o w s from (2) t h a t ( x f - x j : (y # - y u ) : (z, - z J - (x - x / ) : (y - y, ) : (z ~ z,)\u00C2\u00BB and from ( 5 ) we get axx, + byy, + czz, + f (y, z + z, y) + g(z,x + x, z) + h(Xj y + y,x) + ux + vy + wz = ax, 1 1 + by, *\" + cz ( u + 2fy, z,+ 2gz tx ( + 2hx,y, + ux, + vy, + wz t. As a consequence of ( 4 ) the r i g h t number of ( 6 ) i s equal to - (ux, + vy, * wz, + d) \u00E2\u0080\u00A2 Therefore ( 6 ) reduces to 40. (7) axx,+ byy, + czz, * f (y,z + z, y) + g(z,x + x, z) + h(x, y + y ; x ) + u(x. + x,) + v(y + y, ) + w(z + z ; ) + d =\u00E2\u0080\u00A2 0.' Formula (7) i s the equation of the po i n t of tangency of the plane (x, , y ( , z,) to the surface ( l ) . (1) 2. C o n d i t i o n that a P o i n t L i e s on the Surfaceg-Let the equation of the point on the surface be (8) r x + sy + t z - 1 = 0. Comparing equations (7) and (8) v;e have ax + hy * gz ( + u hx ( + by, + f z , + v _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ SX = gx * f y -5- cz, + w _/ux * vy, + ws, + dj . t \" - 1 Put each f r a c t i o n equal to - A . Then ax, + hy, + gz, * u + A r = 0, hx, + by, + f z , * v + A s = 0, gx, * fy, + cz,.+ w + A t \u00C2\u00AB= 0, ux, + vy, + v/z, + d - A = 0. We a l s o have rx, + sy, + tz, - 1 = 0. E l i m i n a t i n g x, , y, , z ( , A from the above equations, ne o b t a i n the r e q u i r e d c o n d i t i o n , namely (1) C. Smith \" S o l i d Geometry\", p. 41. 41 a h S u r h b f V s S f c w t u V w a -1 r s t -1 0 which i s the same as ( 9 ) A r % Bs*\"+ C t % 2Fst + 2Gtr + 2Hrs + 2Ur + 2Ts + 2Wt + D = 0 , where A, B, C, e t c . , are- the c o - f a e t o r s of a, b, c, e t c . , r e s p e c t i v e l y , i n the determinant a h g u h b f v g f e w u v w d The r e l a t i o n ( 9 ) i s a c o n d i t i o n t h a t the p o i n t (8) l i e s on the surface ( l ) I n c i d e n t a l l y , ( 9 ) represents a c o n i c o i d i n the C a r t e s i a n system. Hence, f o r a p o i n t to l i e on the surface ( 1 ) , i t must l i e on a conicoid-;' t h a t i s , ( l ) represents a c o n i c o i d i n the planar system of c o B r d i n a t e s . A proof that ( l ) represents a c o n i c o i d w i l l be given i n s e c t i o n 3, where no reference i s made, as above, to C a r t e s i a n c o o r d i n a t e s . (1) For a s i m i l a r d i s c u s s i o n see Snyder and Sisam, pp. I 3 0 , 131. 42* 3\u00C2\u00BB Locus of Middle P o i n t s of a System of P a r a l l e l Chords:-L e t the equation of the surface be ( l ) , and l e t (8) be the equation of any p o i n t on t h i s e o n i c o i d ; r , s, t must s a t i s f y ( 9 ) . Let (10) Jt-x. + my + nz - 1 = 0 be the equation of a p o i n t on a l i n e whose d i r e c t i o n cosines are y\ , 4/\u00C2\u00BB The p o i n t (8) w i l l l i e on t h i s l i n e and be d i s t a n t p from (10) i f s - m <= p^6c , t - n = p-z/ , that i s s i f r = + p A s S = m + P JUL. , (11) v t = m + p 4/ \u00E2\u0080\u00A2 I f we s u b s t i t u t e (11) i n (9) we ob t a i n a quadratic equation i n p, which shows that any given l i n e cuts the surface i n two p o i n t s . I t f o l l o w s that a l l s t r a i g h t l i n e s i n a plane cut the surface i n two p o i n t s , and ther e f o r e a l l plane sections of the surface are conic s e c t i o n s . This i s the d e f i n i t i o n of a e o n i c o i d . We have (12.) p\"\"(A / f + B ^ + C S\ + 2C-\u00C2\u00AB/A+ 2H/1/-) + 2p(AiM + Bm^ _+ Cn F i i A + FmV + GlS + G-nA + H^A+ HmA \u00E2\u0080\u00A2+ IT A + v>_ + ' WT/) * ( A ^ v * Bm% Cn%\"2Fmn * 2Gn/+ 2Him + 2 * 2Vm + 2Wn + D) = 0, where A, B, C, ... , have the same values as i n s e c t i o n 2. I f (10) i s the equation of the middle point of the l i n e , the values of p obtained from (12) must be equal n u m e r i c a l l y but opposite i n s i g n . The c o n d i t i o n f o r t h i s i s t h a t the c o e f f i c i e n t of p equals zero. Hence (13) i ( A A + Eyu + C V ) + m(H A + B^A. + F-*/) * n(G A + Fyu. + 0/ ) * U A \u00E2\u0080\u00A2\u00C2\u00AB- Yyu- + f i / = 0, Therefore the plane whose p o l a r coBrdinates are given by (14) COS oC = COS y_? cos If where p R*~ * s t-QL R + R + S R *~ + U A + A A + EyU. + G V , H A\" + B G A + F/U + c passes P Q R S through the p o i n t ( l O ) l 1 ^ But (14) represents a (1) C.f. equation (14), Chapter I . 4 4 . f i x e d plane when y\ , /b- , */ are f i x e d . Therefore the mid-p o i n t s of a l l p a r a l l e l chords whose d i r e c t i o n cosines are A * ^ * ^ H e In the plane (14). A plane which passes through the mid.-points of a system of p a r a l l e l chords o f a e o n i c o i d i s known as a di a m e t r a l plane. I f a d i a m e t r a l plane i s perpendicular to the chords i t b i s e c t s , i t i s c a l l e d a p r i n c i p a l plane* 4. The P r i n c i p a l Plane;-I f the plane (14) i s pe r p e n d i c u l a r to the chords whose d i r e c t i o n cosines are A ,yu. , V , the d i r e c t i o n cosines of i t s p o l a r normal must he /\ %^u. , 4/* Therefore AA + H/c * GV H A + B/\"- + i V G A + F>- + c V - \ - ~ ~ Put jT f o r the common value of each of these f r a c t i o n s ; then (A - J ) A + H/c + - 0, (16) H A + (B - 0, GA * F/c + (C - J )fc\u00C2\u00BB z', z = i / , x' + /f/v y' + 4/3 z'. The p o i n t whose equation r e f e r r e d to the o l d system i s - r x + sy + t z - 1 = 0, becomes ( r A + s>/ + t A/, \u00E2\u0080\u00A2)_' * ( r ^ _.+ s/\u00C2\u00ABv + t^ \u00C2\u00ABC )y' + ( r /13 + s/^ + t V, )\u00C2\u00A3' ~ 1 = 0, i n the new system; t h a t i s ' r ' = r A1 + Sju, + t , s' = r ^ v + s ^ + t , t' = r > j + s^k, + t *4 . The in v e r s e r e l a t i o n s are r e a d i l y found to be r = r M , + s' Ax.+ t'Aj s (43) s = 2?/*, + s'/ 3 , t = r 1 ' M u l t i p l y the f i r s t column by , the second by(3 , the t h i r d by Y \u00C2\u00BB a n ( 3- s u b t r a c t t h e i r sum from the l a s t column\u00C2\u00BB In the r e s u l t i n g determinant, m u l t i p l y the f i r s t row by o( the second by , the t h i r d by Y s an3. subtract t h e i r sum from the l a s t row. The r e s u l t i n g determinant i s /\ \u00C2\u00AB Hence , so t h a t i s i n v a r i a n t under both t r a n s l a t i o n and r o t a t i o n . 64 CHAPTER H I C l a s s i f i c a t i o n of Surfaces 1. Review of Previous Work:-In Chapter I I we have seen that the c o n d i t i o n that a p o i n t whose equation i s (1) r x + sy + t z - 1 = 0 l i e s on the surface whose equation i s given by (2) ax'v'+ by1'-!- cz\" + 2fyz + 2gzx + 2hxy + 2ux + 2vy + 2wz +d=0 Is (3) i r r + Bs\"\" + Ct\" + 2Fst + 2Gtr + 2Hrs + 2Ur + 2?s + 2Wt +D\u00C2\u00AB0 where A, B, C, .... , are the c o - f a c t o r s of a, b, c, .... , i n t he determinant a h g u h b f v g f c w u v w d For b r e v i t y we s h a l l r e f e r to (3) as the \u00E2\u0080\u00A2\u00E2\u0080\u00A2point-condition 8 8 equation. We have also seen that 3 = (A + B + C), j = (AB + BC + CA -F\"-G\"\"- H\"), A H G- U H B F \u00C2\u00A5 G F C W TJ Y W D V are i n v a r i a n t under t r a n s l a t i o n and r o t a t i o n . 5\" = A H G H B F G F C and 2. The Sphere;-The sphere i s defined to he the locus of a p o i n t which moves so as to remain at a constant distance from a f i x e d p o i n t . This distance i s known as the radius and the f i x e d p o i n t i s the center of the sphere. Let the equation of the center he c^x + {2> y + z - 1 = 0, and l e t the radius he R; then we have / ( r -d. (s ~ 7 * r + ( t - * T - R, or ( r - . * 0 % (s - y 3 ) % ( t - y ) \" - R.\ Therefore the general p o i n t - c o n d i t i o n equation of a sphere A r % A s % A t % \u00C2\u00A3Ur + 2Ys + 2Wt * D = 0, where A i s d i f f e r e n t from z e r o . Conversely, any poin t r x + sy + t z k 1, where r , s, t s a t i s f y the c o n d i t i o n equa-t i o n , l i e s on a sphere. The p o i n t - c o n d i t i o n equation of a sphere whose center i s the o r i g i n , i s seen to he T * S + t = R e The sphere may a l s o be\"defined as the envelope of planes which move so as always to remain a t a constant d i s -tance from a f i x e d p o i n t . Thus y x N - yN- Z\"1- 1 . oc x + /? y + y z-1 R t h a t i s (4) R \ x % y % 0 = ( ^ x + ^ y + y z - 1 ) \" . The equation o f a sphere, center at the o r i g i n , i s seen to he R\"^\"* y~ + z\"\") \u00C2\u00AB 1, or (5) ax v+ a y % as v+ d = 0. ( I f a and d have the same sign the sphere i s imaginary.) The p o i n t - c o n d i t i o n equation of the sphere (5) i s Ar% - BS^-J- Ct1'-*- 2Fst + 2Gtr + 2Hrs + 2Ur + 2Vs > 2Wt + S = 0, ' where A = a^~d, B a^d, C = a v 4 ( D = a 3 , and F = G = H - T T - V \u00C2\u00AB = W = 0 . Therefore 3 - 3 a ld, o^ ) = a'd 3, ^ = a fd J\u00C2\u00BB 3 . The E l l i p s o i d ; -Consider the surface whose equation i s (6) a x + b y + c z = 1. The p o i n t - c o n d i t i o n equation of t h i s surface i s found to he ( 7 ) b c r + c a s + a b t = a b c . For a, b, c are a l l d i f f e r e n t from zero, and a, b, e i n 67. descending order of magnitude, we have a _ a1\" _ a and Hence a p o i n t on the surface can not be at a distance from the o r i g i n greater than a nor l e s s than c. The surface i s therefore l i m i t e d i n every d i r e c t i o n ; and, s i n c e a l l plane s e c t i o n s of a c o n i c o i d are c o n i e s , i t f o l l o w s that a l l plane s e c t i o n s of (6) are e l l i p s e s . This i s the u s u a l d e f i n i t i o n of an ellipsoid\u00C2\u00A9 The surface i s c l e a r l y symmetrical w i t h respect to the three coordinate planes, the three coordinate axes, and the o r i g i n . The p o i n t s i n which i t cuts the axes are found by l e t t i n g s = t = 0, t = r - = 0, r = s = 0, r e s p e c t i v e l y , i n equation (7)\u00E2\u0080\u00A2 These p o i n t s are determined by the r e l a t i o n s r = \u00C2\u00B1 a, \u00C2\u00B1ax - 1 = 0 , s = _r b, \u00C2\u00B1 by - 1 = 0 , t = \u00C2\u00B1 c, \u00C2\u00B1cz - 1 B 0, r e s p e c t i v e l y . Consider the system o f tangent planes through the p o i n t (8) mz - 1 = 0, on the Z a x i s . The coordinates of a l l planes through t h i s 68. point and touching the surface are (x, J, 1 ) where m a x + b y + e_ = i . -The p o l a r plane of the p o i n t (8) i s (0, 0, m_\ c. Translate c1-the o r i g i n to the p o i n t 2l z - i = or m the new XY plane w i l l be the p o l a r of the p o i n t (8). The equation of (8) becomes ______-\u00E2\u0080\u009E z - 1 = 0; m that i s , the coordinates of a l l planes through (8) w i l l be (V v m v \u00C2\u00BB ^ * T ~y\u00C2\u00BB Let these planes touch the surface whose new m - e equation i s a^x\"1* b^ y\"*\"* c\" z*~ = QL. z + l ] ; so t h a t (9) a x + b y ^ = - v \u00C2\u00AB= . \u00E2\u0080\u0094 m ~ e ^ \u00E2\u0080\u009E m \" (1) Therefore we have an e l l i p s e , ^ ' For m > c the e l l i p s e i s r e a l , and f o r m < c i t i s imaginary. The r a t i o of the semi-axes remains constant, namely a : b. The major semi-axis i s equal to a / 1 - \u00E2\u0080\u0094 , which i s seen t o be zero f o r m = c (1) Valgardsson \"Line Coordinates\"\u00E2\u0080\u00A2 and equal to a f o r m i n f i n i t e l y l a r g e * As m becomes i n d e f i -n i t e l y l a r g e the p o l a r plane (0, 0, m_ ) approaches c o i n c i -c dence w i t h the XY plane. I n the same way we could show that the s e c t i o n of the surface made by the YZ plane i s an e l l i p s e of semi axes b and c and t h a t the s e c t i o n made by the ZX plane i s am e l l i p s e of semi-axes c, a. We c a l l a, b, c the \"semi-axes\" of the e l l i p s o i d . I f a ~ b, the s e c t i o n s p a r a l l e l to the XY plane are c i r c l e s and the surface i s a surface of r e v o l u -t i o n . I f a = b = c we have a sphere. For the e l l i p s o i d Jr = - ( a b + b c + c a ; \ = (a\"bV ) (a\"+ b^+ c*\") ' & - - a'^e^. I f c = 0, (6) becomes a x + b j - 1, and the p o i n t - c o n d i t i o n equation (?) becomes a^b^t 1\" = 0. I f a, b are d i f f e r e n t from zero, then t = 0. Hence f o r c = 0, the surface must l i e wholly i n the XY plane* In t h i s case S ~ - a\"1\" b \ }- \u00C2\u00B0-P - o, A - o. 7 0 . I f b = c = 0, (6) becomes a X l e Hence the surface has degenerated i n t o the two points ax t1 = 0. Let a \u00C2\u00AB= a ( A , b = h, A , c \u00C2\u00AB e, A . Equation (6) then becomes (10) a^x^ b^ y w + \u00C2\u00B0 Let A increase i n d e f i n i t e l y but l e t a, , b ( , c, remain f i x e d . I n the l i m i t vie have a ( x * b, y -s- e, z ^ 0. Hence t h i s equation i s the l i m i t i n g case of an e l l i p s o i d as the semi -axes a, b, c become i n f i n i t e l y l a r g e . I t i s to be n o t i c e d t h a t t r a n s l a t i o n does not a f f e c t the l a t t e r equa-t i o n . Th.et only plane which i s tangent t o the surface i s the plane (0, 0, 0). The p o i n t - c o n d i t i o n equation of (10) I s b e _ c a, - a b v \u00E2\u0080\u009E ' 1 r J L s / _J L t \u00C2\u00AB a b e, . A- A\" K In the l i m i t , when A becomes i n f i n i t e l y l a r g e , t h i s equa-t i o n becomes Or'\" + 0s*~ \u00E2\u0080\u00A2* 0t*~ = a/ b\", , whioh can be s a t i s f i e d only by p o i n t s at i n f i n i t y . I n t h i s case - ^ a ~ /S 0. 71 The Hyperboloid of One Sheet:-(11) Consider the surface whose equation i s a x + b y - c z = 1. The p o i n t - c o n d i t i o n equation of t h i s surface i s found to be (12) D c r + c a s - a b f = a b e. Let a, b, e be a l l d i f f e r e n t ' from zero. The s u r -face i s c l e a r l y symmetrical w i t h respect to the coordinate planes\u00E2\u0080\u009E coordinate axes, and the o r i g i n . By the same method as employed i n S e c t i o n 3, we can show that the plane s e c t i o n s of the surface p a r a l l e l to the XY plane are e l l i p s e s whose axes have minimum values i n the XY plane s e c t i o n , and increase i n d e f i n i t e l y as the s e c t i o n i s moved f u r t h e r away from the XY plane. Thus i s the equation of the e l l i p s e when the plane passes through the p o i n t m The semi-axes are i n the r a t i o a : b and the semi-major a x i s has the value a / m\"+ c f which becomes i n f i n i t e l y m l a r g e as m approaches zero. In the same way we f i n d that sections p a r a l l e l to the YZ plane are hyperbolas. In p a r t i c u l a r , i f we consider the s e c t i o n made by the p l a n e / \u00E2\u0080\u0094 , 0, 0 ) , we o b t a i n the a x + b y = m + c \u00E2\u0080\u0094- ,2, 23 ** 1 \u00E2\u0080\u0094 0 & 72. equation b y - e z = m a*\" This curve i s w e l l - d e f i n e d except f o r m = a, and t h i s i s seen to be the case where the plane i s a t a distance from the YZ plane equal to the semi-axis a of the e l l i p s e which i s formed by the i n t e r s e c t i o n of the surface by the XT plane. We can d i s c u s s t h i s case e a s i e r w i t h reference t o the p o i n t -condi t i o n equation which i s c a s - a b t = a b c - b c r . When r = a we have b c tha t i s \u00C2\u00A3 _ + b_ t c The system of p o i n t s whose equations are ( 1 3 ) t y + t z - 1 \u00C2\u00BB 0 c and (14) - A ty + tz - 1 = 0 c can be shown to define two l i n e s . For the d i r e c t i o n cosines of the l i n e j o i n i n g (13) to the o r i g i n ( S e c t i o n 21, Chap. I ) are cos oC - 0, 73\u00C2\u00BB cos = b , ^ a*\". I f r = a , the e l l i p s e s degenerate i n t o p o i n t s on trie X a x i s . For t h i s s u r f a c e J)- = (GU&L~ + a1\" - t^cT) ^ = S^\TG' (a\"- b\"\"- c\") \u00C2\u00A3 = - a'b'e* = - a ' b V . When b or c i s zero, cases are obtained which have been discussed a l r e a d y . Let us consider the case when the semi-axes become i n f i n i t e ; suppose the equation i s a x - b y - c z = \u00E2\u0080\u0094 A Then there i s no p a r t of the surface between the planes p a r a l l e l to the YZ plane and passing through the p o i n t s \u00C2\u00B1 a A x - 1 = 0. I f A approaches i n f i n i t y the distance between these points becomes i n f i n i t e . In the l i m i t we have the hyperboleid of 76, two sheets at i n f i n i t y . We have J = ^ ==<^ -/\== 0. 6. She P a r a b o l o i d ; -Consider the surface defined by the equation ( 1 7 ) b^ y*\" i- c's 1\" + 2ux - 0 . The p o i n t - c o n d i t i o n equation of ( 1 7 ) i s ( 1 8 ) G^VL'S\" * b ^ u ^ t + 2b\"c\"ur ~ 0. I f b, c, u are a l l d i f f e r e n t from aero, we may w r i t e , i n s t e a d of (18), . \u00C2\u00A3 . + 0 . b\" _ c\"~ u The surface ( 1 7 ) I s symmetrical w i t h respect to the XY and ZX planes and the X a x i s . The p o l a r of the p o i n t ( 1 9 ) mx - 1 \u00C2\u00BB 0 i s ( S e c t i o n 1 0 , Chap. I I ) the plane / , 0 , 0 j\u00C2\u00AB Translate the o r i g i n to the p o i n t - mx - 1 \u00C2\u00AB 0o Then the p o l a r plane w i l l be the new XY plane., Equations ( 1 9 ) and ( 1 7 ) , r e f e r r e d to the new axes, are r e s p e c t i v e l y 2mx - 1 = 0 , b^y\" * ^ z 1 \" - 2umx*~ + 2ux = 0 . l e t a l l the tangent planes pass through the p o i n t ( l ? ) ; t h a t i s x = 1 _ . Therefore we have 2m ^ u b y~ + c\"2 ^ B - 2m \u00E2\u0080\u00A2 Hence plane s e c t i o n s p a r a l l e l to the YZ plane are e l l i p s e s 77-of semi-axes J-2m and c tt 2m \u00C2\u00AB This e l l i p s e degenerat XL es to a p o i n t when m = 0; that i s , the YZ plane touches the surface at the o r i g i n . The e l l i p s e increases i n s i z e as the c u t t i n g plane i s moved f u r t h e r from the o r i g i n . I t i s to he noted that m and u must be opposite i n s i g n f o r r e a l e l l i p s e s . I f u I s p o s i t i v e the surface l i e s w h o l l y on the p o s i t i v e side of the YZ plane*, Consider any plane p a r a l l e l to the JZ plane, (0, m, 0 ) , say. Trans l a t e the o r i g i n to the p o i n t \u00E2\u0080\u00A2L y - 1 \u00C2\u00AB 0 ; m that i s the new XZ plane i s t h i s plane. Equation ( l 8 ) becomes (Chapter I I ) For any p o i n t i n the new XZ plane S = 0 . Therefore the p o i n t - c o n d i t i o n equation o f the plane s e c t i o n by the new JZ plane becomes ( 2 0 ) t 2r + 1 c XL m\"b I t can eas i l y be shown the l i n e - c o n d i t i o n equation f o r a parabola has the same form as ( 2 0 ). Therefore the ( l ) This can be dose by a method s i m i l a r to that employed i n Chapter I I , S e c t i o n 2 . See Snyder and Sisam, p. 9 1 . 78 s e c t i o n by t h i s plane i s a parabola,, I n the same way we can show that sections p a r a l l e l t o the XY plane y i e l d parabolas\u00E2\u0080\u00A2 We c a l l the surface whose equation i s (17) an e l l i p t i c p a r a b o l o i d , because the s e c t i o n s p a r a l l e l to one c o o r d i -nate plane are e l l i p s e s and the sections p a r a l l e l t o the other two c oBrdinate planes are parabolas. In the same way we can i n v e s t i g a t e the surface whose equation i s (\u00C2\u00A31) b v y - c v i ' ' + 2ux = 0. Sections p a r a l l e l to the YZ plane y i e l d hyperbolas and sec-t i o n s p a r a l l e l to the other two coordinate planes y i e l d parabolas,. Therefore (21) represents an h y p e r b o l i c para-boloid\u00C2\u00AE For ( I ? ) J - ~ u v ( b % - O , j - b ' o V ; &\u00E2\u0080\u00A2 \u00C2\u00BB 0, /\ <= b' GC u 6 . For (21) J \u00C2\u00AB - u\"(b\"- ou), ^ = - b ^ u * , ^ - 0, A - - b* c 4u\" . I f u = 0, we have b y + c z =0 or b y - c s =0. The f i r s t i s a s p e c i a l case of the i n f i n i t e e l l i p s o i d , and the second represents a p a i r of i n f i n i t e l y d i s t a n t p o i n t s F o r these two cases - ^ - = = 0o When e = 0, we have b^y \" + 2ux = 0. This i s a parabola i n the XY p l a n e I n t h i s ease 3- = - b ? u u , jj< = = A \u00E2\u0080\u00A2= 0. The p o i n t - c o n d i t i o n equation reduces to that i s , t = 0, and the p o i n t s a l l l i e i n the XY plane. (3) 7. I n v a r i a n t s f o r the Various Equations;-Equation A a vx\"+ b > ^ c \" z v = 1 - _ + a x + b y - c z = l + ? ? a x - b y ~ c z < = l - - ? ? b^y^- c^z'-s- 2ux \u00C2\u00AB 0 - 0 + _ b^y\"- c\"z\"l\"+ 2ux = 0 + 0 ? (1) Valgardsson \" l i n e GoBrdinates\", Ch. I I I . (2) Valgardsson, Ch. I I , Sect. 4. (3) I t i s understood that a l l c o e f f i c i e n t s appearing i n the f o l l o w i n g t a b l e are d i f f e r e n t from zero. 8o, Equation A 3- 3 a x v + by\"1' + ess1\" = 0 0 0 0 0 a x v + b y \" = 0 0 0 0 0 a Tx v+ b V = 1 0 .0 0 \u00E2\u0080\u00A2* a x - b y <= 1 0 0 0 b\"y%- 2ux = 0 0 0 0 \u00E2\u0080\u00A2f 8, X ~ X 0 0 0 0 Since these are a l l the p o s s i b l e equations, we can say that when. \u00C2\u00A3 ^ 0, \u00C2\u00A3 0 we have an e l l i p s o i d or an hyper-b o l o i d . of one or tw o sheets. If A ^ 0, \u00C2\u00AB\u00C2\u00AE = 0 we must have e i t h e r an e l l i p t i c or hype r b o l i c p a r a b o l o i d . I f ^ = =. jj - 0, and J? =f^ 0 we have a plane curve, which can be an e l l i p s e , parabola, or hyperbola. I f A - = ^ *= _9 ^ 0, the e quation represents two p o i n t s , or else may be s a t i s f i e d o nly by po i n t s a t i n f i n i t y . The o r i g i n a l equation represents two po i n t s when i t has two l i n e a r f a c t o r s i n x, y, z, f o r which a necessary c o n d i t i o n i s t h a t the d i s c r i m i n a n t & vanish. 81 CHAPTER IT Reduction of the General Equation 1. General Statement In t h i s chapter we s h a l l consider the r e d u c t i o n o f the general equation when A G, that i s , when the equation represents an e l l i p s o i d , hyperboloid, or parab o l o i d * 2. Reduction of the P o i n t - C o n d i t i o n Equation;-Let the equation Ar\"+ BS^-J- Ct\"+ 2Fst + 2Gtr + 2Hrs * 2Ur + 2Ys (1) \" ' + 2Wt *\" D = 0 be the point-condi t l o n equat ion of the surface ax\"* by\"*'+ cz1'-!- 2fyz + 2gzx + 2hxy + 2ux + 2vy (2) \" \" * 2wz +\"d \u00C2\u00AB 0. We have seen ( S e c t i o n 4, Chap. I I ) that there i s at l e a s t one p r i n c i p a l plane. Take thi s plane f o r the XY plane i n a new system of co o r d i n a t e s . The degree of ( l ) w i l l be una l t e r e d by the tr a n s f o r m a t i o n . By s u p p o s i t i o n the XY plane b i s e c t s a l l chords p a r a l l e l t o the Z a x i s j t herefore i f r ( x + s ( y * t ( z - l = 0 be any p o i n t on the surface, the p o i n t r ( x * s y - t, z - 1 = 0 w i l l a l so be on the surface. From t h i s we see t h a t i n the transformed equation F = G- = W = 0. The reduced equation therefore i s A r % Bs\" + Ct\"+ 2Hrs + 2Ur + 27s + D = 0. l\Tow r o t a t e the X, Y axes through an angle O g i v e n by the r e l a t i o n tan 2& = 2H . A-vB according to the transformations (4j>) of Chapter I I , namely r = r'oos & + s ' s i n Q , s = - r' s i n 0 + s' cos 6 , t \u00C2\u00AB t ' . ; Dropping primes, we get an equation of the form O ) A r % Bs w+ Ct v+ 2Ur + 2Ys + D = 0\u00C2\u00BB ( i ) Let A, B, C be a i l f i n i t e and d i f f e r e n t from zero. We can then w r i t e equation (j>) i n the form A ( r * TJ ) \ B/s + T ) \ Ct\"= U~4 \u00C2\u00A5\"~ DHD'. V A / V. B\"7 A . B\" Hence, by changing the o r i g i n to the p o i n t | x + J y - 1 - 0 by means o f formulae (46) of Chapter I I , we obta i n Ar\"\" + Bs\" + C t \" \u00C2\u00BB 3)'. I f D1 be not zero we have 83. which we can w r i t e i n the form (4) a b c or (5) r \" s ^ \u00E2\u0080\u009E t a b c or (6) r 1 \" s\"*- t E 1, ac cor ding as D; , D' , D' are a l l p o s i t i v e , two p o s i t i v e and A 1\" U one negative, or one p o s i t i v e and two negative, r e s p e c t i v e l y * ( i f a l l three are negative the surface i s c l e a r l y imaginary.) I f 3' be aero, we have (7) A r \" + Bs' u -s- C t \" =0. ( i i ) Let A, any one o f the c o e f f i c i e n t s , he zero. Write the equation i n the form 2Ur + B^s + Y y C t V + D - Y v = 0. I f U be not zero, by changing the o r i g i n t o the p o i n t Qx + TJ y -* 1 = 0, whe re Q, = 1 / D - T , , 2tr we can reduce the equation to (8) B s \" + C t \" + 2Ur = 0, 8 4 . I f U = 0, we have the form (?) B s \" + C t v + D' = 0, or, i f I)'= 0, the form (10) Bs\" + Ct\" = 0. ( i i i ) L e t B, C, two \u00C2\u00A9f the three c o e f f i c i e n t s , be zero. We then have A^r + ] J j \ ZJa * D' -I f we t r a n s l a t e the o r i g i n to the p o i n t U - 0, A the equation reduces to the form (11) r.*\" = 23ts. I f , however, \u00C2\u00A5 = 0, the equation i s equivalent to (12) r*\" = 3c' \u00C2\u00A9 3 o go Find the Equations of the Center of a Conicoid.:-I f the o r i g i n i s the center of the s u r f a c e , i t i s the middle p o i n t of a l l chords passing through i t; i f r , 2 c - t - s 1 y * t | z - l = 0 be any p o i n t on the surfac e , the point - r ( x - s ( y - t, z - 1 = 0 w i l l also be on the surface\u00C2\u00BB Hence we have Ar,\" + Bs,\" * C t ^ + 2Fs, t, * 2Gt, r ( + 2Hr( s, + 2Ur + 2Vs, ~ ' \" \" \u00E2\u0080\u00A2'\u00E2\u0080\u00A2 2wt, + D \u00C2\u00AB o; 85 and Ar,\" + Bs^ + Ct, v + 2Fs, t, + 2Gt, r, + 2Hr, s, - 2Ur, the r e f o r e - 27a, -\u00E2\u0080\u00A2 2Wt( + D \u00C2\u00AB 0; TJr Vs ; + Wt, \u00C2\u00AB= 0. Since t h i s r e l a t i o n holds f o r a l l p o i n t s on the surface, we must have U, V, W a l l zero. Hence, when the o r i g i n i s the center o f a c o n i c o i d , the c o e f f i c i e n t s of r , s, t are a l l zero. Let o r x + ^ y + y z - l \u00C2\u00AB = 0 be the equation of the center of the s u r f a c e ; then i f we take.the center f o r o r i g i n , the c o e f f i c i e n t s of r , s, t i n the transformed equation w i l l a l l be zero. The transformed equation w i l l be ( S e c t i o n 46, Chap. I I ) A ( r V \u00C2\u00AB< ) V + B(s + ^ f + C(t + / ) x+ 2F(s + ^ ) ( t i \u00C2\u00ABf ) + 2G(t + y ) ( r + a<) + 2H(r + ^ \" ) ( s +^ ) + 2tJ(r + <<) * 27(s + /3 ) + 2W(t + ) + D \u00C2\u00AB 0. Hence\"the equations g i v i n g the center are Ae(. * H ^ + G T + IT \u00C2\u00AB 0, H<< + B /3 + F 2T + 7 = 0, G *c + Fy3 + C r + W \u00E2\u0080\u00A2\u00C2\u00BB 0. (13) The r e f ore (13a) H G TJ A \u00E2\u0080\u00A2 G U A H TJ A H G B F \u00C2\u00A5 H F T H B t. H B F F C G C \u00C2\u00A5 G F W G F C Q f The p o i n t - c o n d i t i o n equation of the e o n i c o i d when the center i s a t the o r i g i n i s (14) Ar\"+ Bs v+ Ct\"+ 2Fst + 2Gtr + 2Hrs + D ' = 0, where D ' i s obtained from (3) by p u t t i n g r = <*\" , s -/3 , t = tY \u00E2\u0080\u00A2 M u l t i p l y equations (13) i n order by \u00E2\u0080\u00A2< , ^ , % and s u b t r a c t the sum from D j then we have. (13) V = TJe< + Y/3 + W * + D* From (13) and (15) we have A H G TJ H B F Y I \u00C2\u00BB 0; G F G W TJ:- Y w D-D'! therefore (16) D' A H G- ,A H G IT H B F S B \u00E2\u0080\u00A20 7 G F ,c G F\" 0 W U\" Y w D which may be w r i t t e n (17) v'c\u00C2\u00A3> \u00C2\u00AB A . I t i s seen that the equation of the center i s gi v e n by (18) x + V y + s - \u00C2\u00A3> - 0, where e t c . , are the co-f a c t o r s of IT, Y, e t c , In Z\ 87. 4. The D i s c r i m i n a t i n g Cubic;-We have seen ( S e c t i o n 2) that by a proper choice of rec t a n g u l a r axes Ar*+ Bs^* Gt%- 2Fst + 2Gtr + 2Hrs can always be reduced to the form c< r v+ ^ s\"\"+ y t - 1\"; and t h i s r e d u c t i o n can be e f f e c t e d without changing the o r i g i n , f o r the terms of second degree are not a l t e r e d by transforming to any p a r a l l e l axes. How r v + s^+ t *Is u n a l t e r e d by a change of rectangu-l a r axes through the same o r i g i n . Hence, when the axes are so changed t h a t A r v + Bs^+ C t % 2Fst + 2Gtr ->- 2Hrs becomes r^+ (h s^+ V t \ (19) Ar^+ Bs^+ GtN- 2Fst +' 2Gtr + 2Hrs - J ( r \" + s\" + t\" ) w i l l become (20) ^ r v + ^3 s % f t * \" - J'(rx'+ a*\"* t . Both these expressions w i l l t herefore be the pro-duct of l i n e a r f a c t o r s f o r the same values of J \u00E2\u0080\u00A2 The c o n d i t i o n that (19) i s the product of l i n e a r f a c t o r s i s (21) | A - J H G II B - f F 1=0, G F C -jT But (20) i s the product o f l i n e a r f a c t o r s when J i s 88. equal to oC , ^ , or , Hence oc , p s are the three r o o t s of (21). The equation when expanded i s f* - f (A + 3 + C) + f (AE + BC + CA - F*~- G-*\"~ H*) - (ABC + 2FGH - A3?*\"- SG^-CH\") = 0, or a s ) y -&r*w - -This equation i s c a l l e d the \" d i s c r i m i n a t i n g c u b i c \" . 5\" D i s c u s s i o n f o r JQ \u00C2\u00A3 0:-From equation (18) we see tha t there i s a d e f i n i t e center a t a f i n i t e d i s t a n c e , unless \u00C2\u00AB= 0. I f 0 and one o f ^ , s % V t + 3)' = 0. Then, by Section 3, c*T , /3 , Y w i l l be the three r o o t s of the d i s c r i m i n a t i n g c u b i c . Since D'- /\ , the l a s t equation may be w r i t t e n i n the form JiU r % &/!> s\"* ^ / t \" + Z\ =0. I f the three q u a n t i t i e s \"P* , 'QJJL. , \u00C2\u00B0^J^ are 8?. a l l n e g a t i v e , the surface i s an e l l i p s o i d ; i f two of them are negative, the surface i s an hyperboloid of one sheet; i f one i s negative, the surface i s an hyperboloid of two sheets; and i f they are a l l p o s i t i v e , the surface i s an imaginary e l l i p s o i d . We have shown i n Chapter I I that the general equation can be reduced t o one of the three forms (23) azv+ by* + oz\" - 1 = 0, (24) ax\"+ by\"-*- c s w ~ 0, (23) b y % cz*\" + 2ux*= 0* We see from Section 7\"of Chapter I I I that \u00C2\u00A3 0 always r e q u i r e s A -fc 0, which i s true only f o r (23)\u00C2\u00AB. b. D i s c u s s i o n of the Case j9 = 0 : -When = 0, one root of the d i s c r i m i n a t i n g cubic must be zero. From Se c t i o n 4, Chapter I I , we see that one p r i n c i p a l plane must be the plane (0, 0, 0)\u00C2\u00AB I f 0, vie must have tw0 f i n i t e p r i n c i p a l planes, and therefore the center i s a t i n f i n i t y and must l i e on the l i n e of i n t e r -s e c t i o n of the two f i n i t e princ i p a l planes\u00C2\u00A9 I f $ - 0 and Z\ 0, equation (18) shows that the center i s a t i n f i n i t y . Since one root of the d i s c r i m i n a t i n g cubic i s zero, the equation can e a s i l y be solved; l e t the root s be 0, oC > ^ 3 . Find the d i r e c t i o n cosines of the p r i n c i p a l a x i s by means of equations (16), Chapter I I , and take the Z a x i s p a r a l l e l to Hie p r i n c i p a l a x i s * The 90, equation w i l l then become v+ (t> t* + 2U'r + 2V's + ZW't + JD = 0 , or, by s change* o f o r i g i n , \u00C2\u00AB=<\u00E2\u0080\u00A2 s v + t % - 2U 'r = 0 . Hence we have the su r f a c e , which, expressed I n plane coordinates, i s ay v+ bs^* 2ux = O,^1^ since A ^ 0 . 7 . Summary;-Let us i n v e s t i g a t e the general equation of a e o n i c o i d . I f A ^ 0 and d 7^ 0 , i t f o l l o w s that JQ \u00C2\u00A3 0 and w e have an e l l i p s o i d , or hy p e r b o l o i d . I f & i s p o s i t i v e we have the hyperboloid of one sheet. I f & i s negative we d i s c o v e r the nature of the surface by s o l v i n g the d i s c r i m i n a t i n g cubic-three r o o t s w i t h the same sig n denote an e l l i p s o i d and r o o t s which d i f f e r i n s i g n denote an hyperboloid o f two sheets,, I f Z\ \u00C2\u00A3 0 but d = 0 , i t f o l l o w s thatj@ = 0 \u00E2\u0080\u009E ^ This gives us an e l l i p t i c or h y p e r b o l i c p a r a b o l o i d according as A i s negative or p o s i t i v e , r e s p e c t i v e l y . The plane curves are found to be those surfaces f o r which a l l the i n v a r i a n t s except 3 vanish. I f d = 0 the plane curve i s a parabola. I f d / 0 the plane curve i s an (1) Snyder and Sisam, p. I 5 0 . ( 2 ) S e c t i o n 7, Chapter I I I . 91 e l l i p s e or hyperbola according as J) i s negative or p o s i t i v e , r e s p e c t i v e l y * A p a i r o f p o i n t s i s g i v e n Mien = ^ ~ = \u00C2\u00A3\u00C2\u00B1 ~ 0 provided t h a t the equation i s f a c t o r a b l e e Otherwise the equation represents an i n f i n i t e c o n i -c o i d or an i n f i n i t e conic * BIBLIOGRAPHY 1. Lambert \" A n a l y t i c Geometry 1 3, The Macmillan Co., 1904. 2. Smith \" S o l i d Geometry\", Macmillan and Co., 188?o 3c Snyder and Sisam \" A n a l y t i c Geometry of Space\", Henry H o l t and Co.,1 1914. 4. Tanner and A l l e n \" B r i e f Course i n A n a l y t i c Geometry\",' American Book Co., 1911. 5. valgardsson \"Line Coordinates\", M.A. Thesis a t U n i v e r s i t y o f Manitoba. 6. Wilson \" S o l i d Geometry and Conic S e c t i o n s \" , Macmillan and Co., 1898. "@en . "Thesis/Dissertation"@en . "10.14288/1.0080546"@en . "eng"@en . "Mathematics"@en . "Vancouver : University of British Columbia Library"@en . "University of British Columbia"@en . "For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use."@en . "Graduate"@en . "Plane coordinates"@en . "Text"@en . "http://hdl.handle.net/2429/38465"@en .