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An illustrative example of asymptotic oscillations within the helium atom Kadzielawa, Joseph Leon 1938

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An I l l u s t r a t i v e Example of Asymptotic O s c i l l a t i o n s within the Helium Atom. by Joseph Leon Kadzielawa A Thesis submitted for the Degree of MASTER OF ARTS in the Department of MATHEMATICS THE UNIVERSITY.OP BRITISH COLUMBIA A p r i l , 1938 Contents: 1. Introduction page 1. 2. The D i f f e r e n t i a l Equations 2. 3. Lagrange's Quintic -.2. 4. The Equations of Variation .... ...4. 5. The Characteristic Equation 7. 6. The Asymptotic Solutions 13. 7. The Numerical and I l l u s t r a t i v e Example 13. 8. References 24. 1. An I l l u s t r a t i v e Example of Asymptotic O s c i l l a t i o n s  within the Helium Atom. Introduction: The object of this paper is to construct an i l l u s t r a -tive example of asymptotic orbits for the neutral helium atom. The system consists of two electrons and a central body ca l l e d the nucleus. There are forces of at t r a c t i o n and repul-sion between the bodies due to g r a v i t a t i o n a l and e l e c t r i c a l forces. These obey the inverse square law. (1) W. E. Cox by neglecting the g r a v i t a t i o n a l a t t r a c t i o n between the masses determined iagrange's q u i n t i c . This was (2) extended by H. E. Buchanan in his discussion of the o s c i l -(3) lations near the equilibrium points. Miss A. Rayl derived Lagrange's quintic from which the equations of var i a t i o n were obtained by considering also the gr a v i t a t i o n a l forces. This' (4) was extended by Dr. D. Buchanan to the determination of the orbits.along which the three bodies approach their equilibrium positions when given an i n i t i a l displacement from these points. Before the numerical.example is: determined, the constants in the equations of va r i a t i o n and. the solutions of the charac-t e r i s t i c equation w i l l be found using the fact that- the charges and masses of the two electrons are equal. This gives some very useful relations and hence s i m p l i f i e s the computation. 2. The D i f f e r e n t i a l Equations; Let the masses of the two electrons and nucleus "be de-noted by m-p m3, and mg. and carry the charges -e^, - e 3 , and. eg respectively. Let (f^-*^^) represent the coordinates of m^  referred to the centre of gravity as o r i g i n and t h e a x i s rotating about the f- a x i s with uniform angular v e l o c i t y <-0 . Let r. . be the distance between m. and m... Then the d i f f e r e n -t i a l equations derived from Newton's and Coulomb's laws are cit 2 p where and kg*" are factors of proportionality. Tha units o p of mass and of charge may be so chosen that k^<& kg =1. Lagrange's Q.uintic; We w i l l f i r s t give a b r i e f outline of Miss Rayl's deter-mination of Lagrange's qu i n t i c . We aim to f i n d the solution of (1) such that the three masses l i e i n a straight line and are moving in c i r c l e s about the o r i g i n with angular velocity 1*-'. The coordinates are then constants and</ df* di Consider f i r s t the case in which the nucleus l i e s between the electrons. Choose ij^/*. f , and the unit of distance so that r-^g- ^-j - 1. Equation (l) then becomes (2) (a) 0>) 3. (c) I /yrti<o /i -<- _ G <- ? ~ C where a l g - + m-jm2, ai3"= e l e 3 " mi m3» a23 ~ ^ 2 e3 • m2 m3* The second of these equations may he replaced by the center of gravity equation (3) /** r , / , (f, -<-i) f3 O which may be obtained by adding the equations in (2). Set A= f3-fz . Then from (2,a) and (3) £ - - m2 t "13(1 4- A) where M •= M-, +m2 +1113 J / - M Eliminate ^ from (2,a), (2,b) and (3). By making use of the substitution for Lagrange's quintic i s obtained. m 3 a 1 2 ( m l * m 2 ) A m 3 a 1 2 ^ m l 2m2)A -V m3\a-i2(3m;i+m2 ) (4) - 1 1 1 ^ 2 3 - m 2 a 1 3 J A 3 - mija 2 3(3m 3+m 2) -m 3a 1 2 - m 2 a 1 3 j A 2 - mia 2 3(3m 3 + 2m2 )A - m^a23(m2 + m 2)=0. 2 o At least one r e a l positive root exists i f a^2m2 > a13 m3 • For the neutral helium atom l e t m = m-^  - m^ ; 2e = e 2 ~ 2e^-2e 3. Substitute these values i n (4), then (2e 2 t mm2)(m + m 2)A 5 + (2e 2 • mm2)(3m + 2m 2)A 4 • £e 2(4m • m2) • mm22 • 3m2m2 A j^ - je 2(4m + m2) + mm22 * 3m 2m 2jA 2 - (2e 2 • mm2)(3m + 2m2)A - (2e 2 • mm2) (m + m2) s 0. This equation has only one re a l positive root, A = l . If the nucleus does not l i e between the e l e c t r o n s J t 7 j > The unit distance for t h i s i s chosen so that r ^ 3 - ^  - f{ - 1. 4. To f i n d the equation for t h i s case interchange £ 3 , e 3, 1113 and §2* -e2,m2 'respectively, that i s interchange subscripts 2 and .3 without changing the sign of the l e t t e r affected except i n the case of the a's where the sign i s changed with'one change; of the subscript. In t h i s case A s / 2 3 and Lagrange's Quintic becomes: *2 a13^ r al "* m 3 ^ ^ 4 m 2 a 1 3 ^ m l * 2 m 3 ) ^ ^ + m 2 ^ a i 3 ^ m l + m3 ^  + m i a 2 3 - m 3 a 1 2 J A 3 + m 1ja 2 3(3m 2 * m3) + m^a-^U" m 3 a i 2 A 2 m 1a 2 3(3m 2 + 2m3 )A + m^a 2 3(m 2 + m3) = 0. There w i l l exist one r e a l positive root A i f the magnitude of the g r a v i t a t i o n a l a t t r a c t i o n i s greater than-thet r e p e l l i n g forces due to the charges, that i s i f m^m3 > e^e 3. For the normal helium atom mim3 ^ e^e3. Thus we have only to consider the case when the nucleus l i e s between the electrons. The quintic f o r this case gave the value A = l . Let the corresponding coordinates of the three bodies be ( 1 = 1,2,3). ^ ^ z f t ^ 0 j ( i m , ^ J ) m Let a l g a ^ B and a 1 3 a 1 3 _ C The Equations.of Variation: L e t f i -1 i 4 xi» ^ i ' ^ i + y±* i i = ^ i + z i (i =1, 2, 3) where the x^, y^, represent the components of the displacement from the equilibrium points. Substitute (5) in (l) and l e t the variables ^2*^2*^2, D e eliminated by the cSnter of gravity equations. The r e s u l t i n g d i f f e r e n t i a l equa-tions are the equations, of v a r i a t i o n for the straight line solution. They are 5. • i t " a l t (6) (a) (D) where the P^(n)', Q j ^ n ) , R^(n) are polynomials of degree n in the x's, y's, z's developed hy Taylor's formula. Only the f i r s t and second degree terms are. considered. The li n e a r terms are P i U L A n X l t A i 3 x 3 ; q . ( l L D l i y i • D i 3y 3; R i ( l ) = E i l z l " E i 3 z 3 where 11 - A33 = 2(B - C) + 2B 13 O l - — • , 1 m2 m m A,„ .-a A-zn _ 2B + 2C • P contains no l i n e a r term in yj_ m D 11 - D13 - D31 -= D33 - E13 « E31 - ? - £ mg m 2 - 33 - C E l l = E 3 3 r m (1) contains no linear term in x-j From (2a)., (2b) 00 _ B - 2G m m The following are useful in l a t e r computations D l l " ^ + Ell» 11! A w = -2D 11* The second degree terms are T, ( 2 ) 1 -2 2 , 2 1 1 - y,- - z. 2 2. 2 3B(m + m gr - 3C mmQ 6. 6B(m + m ) +• 3C r?™ • 2 + / x l x 3 ' ^ 3 " z l z 3 I 2 ~2 ra2 2m m P (2) / x 2 2 . z 2 V T 2 2. - f x l x 3 " y i v 3 " z l z 3 - 2 2, 2 2 2 2" ~ 3C - 3Bm 2m m^ 6B(m + m2) 3C m. m x 3 " y 3 2 " Z3 '\ J 3G ' 3B(m mnig 2 3C - 3B(m + nig)-' 2m - (x 1y 3+. X g y ^ B U • m2) + 3C 2m X3y3("3G - 3Bm 1 < 2m mr 1 (2) ^3 = x i v i j 3 ^ " 5°.^  * ^ x 3 y i 4 x i y 3 m2* 3B(m • m2) * 3G 2m m22 + x 3 y 3 |3B(m * m ) - 3C 2m mmg (2) (2) (2) 12) R^ ' and R 3 v < i' may be obtained from Q,i and Q,i by re-placing y i by z±. These results may be obtained from those obtained by Dr. D. Buchanan by setting m-^  =• m3 m and A = 1. The Characteristic Equation: The ch a r a c t e r i s t i c equations of (6) are found hy consi-dering only the lin e a r terms. Thus (6,a) gives d 2x 1- 2<^dy1 - A-QX-L - A 1 3 x 3 * 0 d£- dt~ .2. (7) d x 3 - 2wdy3 - A„ X ] L - A 1 ] Lx 3 - 0 d t 2 dt d ^ y i + 2u»dXl - D n y ! - D l xy3 = 0 ""dt^ "IT d 2 y 3 + 2udx 3 - D l l 7 l - D u y 3 dt The c h a r a c t e r i s t i c equation i s 2^ >> 0 (-air - A N ) o 2 * * ^ 0.. D n ( - A n ) A 1 3 Al3 -2co> 0 and this expand.s. into .; -2 u?X D 11 ( -*? - D l l ) - 0 >? + (-2A1X - 2 D n + 8uF) .+ X2(u,4 + 8 i o 2 D i : L - 8 D U + 8 t o 2 ( A 1 1 D 1 1 • A ^ D n ) + 2 ( A 1 3 2 - A n ) 2 D n j - 0 or >2U6 * >v4(u,8+ 2 D 1 1 ) + > | ( J • 8 - 2 D 1 1 - SD^ 2) 2 2 i + 2w(3u> - 4 D 1 1 ) D 1 1 1 = 0. Thus > 2(* 2 * u£)[>v4 4(2D n 4u, 2)"* 2 + 2D 1 1(3«- 2 - 4 D 1 1 ) | - 0 . If >^ and v are the roots of the quadratic in A 2 v . r 4 ; n n 2^ . ™ 2 8. The roots of A are therefore >^ o, o, t w, ^  \Tp , ± . These results are equivalent to those obtained by Miss Rayl. The c h a r a c t e r i s t i c equation for the system (6,b), that is for f d 2 z i - E-QZ-L - E._z^ 0 '13*3 (8) dt' d z 3 " E 1 3 z l " E 1 1 Z 3 ~ 0 1 7 is (9) E 13 E 13 - 0 This gives 4 2>s 2B l x - ET ^ 2 + E-, -i 2 -=• 0 '13 J l l 4 ?v - 2(D 1 X - u / 5 ) ^ - 2D * ^ = 0 (j^+^JO 2 - 2D +u,2) •= 0 Let CT" = f2D 11 Then the roots of (9) are' The solutions of (7) and (8) are therefore (10) f x i - L n ^ 2 t t L i 3 e ^ t ^ i 4 e i ^ L i 5 e ^ * L i 6 ^ \ L i 7 e ' r t + L i 8 i , R ? t y± ~ M i l » M i 2 t + M i 3 e 1 ^ t + M i 4 e i ^ t + M i 5 e ¥ t * M i 6 e ^ * M i 7 e ^ t M i 8 e ^ where the L's, M's^and N's are constants of which eight of the L's and M's and four of the N's are a r b i t r a r y . It w i l l be seen in the numerical example that not a l l the exponents are imaginary. 9. To determine the relations among the constants i t i s necessary to substitute (10) in (7). and (8) and equate^the constant terms, the co e f f i c i e n t s of t and the c o e f f i c i e n t of e* 1"*, e ^ and e^l Then ( L 1 3 i 2 ^ e i w % L 1 4 ^ ^ ( M ^ t M ^ i ^ e ^ - M ^ i M j e ^ - A ^ L ^ L i g t t L - ^ e ^ ^ L ^ e ^ ^ - A ^ L s i t L ^ U L ^ e 1 ^ ^ ^ ^ = 0 ( L ^ i W " ^ ^ ^ - 2 w ( M 3 2 + M 3 3 i " e i w t ^ - A ^ L ^ ^ ^ e ^ ^ ) S 0 ( M 1 3 i 2 u , 2 e i < o t + M 1 4 l 2 M , 2 e ^ M ] . ^ e ^ - M ^ p e ^ ^ / e ^ - M i Q ^ e ^ ) - D i i ( M 3 1 + M 3 2 t + M 3 3 e i w t ^ M 3 4 e i ^ t t M 3 5 e ^ t 4 M 3 6 § t * M 0 ( M g . g i ^ e ^ M ^ i 2 ^ 2 ^ . * M L 3 2 « L 3 3 i - e i - t - L 3 4 i ^ ^ -D 1 1(M 1 1+M 1 2t +M 1 3e 1 " t t M 1 4e i^ ttM 1 -D 1 1(M33,4M 3 2ttMg 3e i u , t +M3 4e^ t^M 3 5e^ ttM3 6e^ t*M3 7e^ t^M 3 8e^ t) ^ 0 is Equate the c o e f f i c i e n t of t to zero. Then A11 L12 + A13 L32 - 0  A13 L12 * A H L 3 2 "= 0 10. Therefore I ^ A - ^ A - Q - A 1 3 A 1 3 ) - 0 Hence L = 0 and also L „ Q = 0 12 «J <s M 1 2 D U + M 3 2 D H ' ° M 1 2 D U • M 3 2 D 1 ; L - 0 Therefore M = - E U C 12 6 d 2u>L12 - D N M N - D n M | ] L - 0 Therefore i f M . ^ i s a r b i t r a r y M 3 1 — From 2 « L 3 2 - D u M n - D n M 3 2 - 0, Mg2 -M12 2 - M 1 2 v A n L n • A 1 3 L 3 1 = 0 2"M32-* A 3 1 L 1 3 _ + A M L 3 1 - 0 give L X 1 = 2 ^ ( - A 1 3 - A l i ) M 1 p _ - 2 M 1 2 A N 2 - A 1 3 2 3 ^ LJ5* 2 U > ^11 • A13>M12 2 M 12 " 2 2 A A 7to 11 A 1 3 6 i c o t Equate the c o e f f i c i e n t s of e to zero " ^ L 1 3 - ~ 2 1 i ° 2 m 1 3 " A n L i 3 " A 1 3 L 3 3 " 0 - c o 2L 3 3 -'21.^3 - A J 3 L 1 3 - A U L 3 3 = 0 - ^ M 1 3 * 2 i ~ L l 3 " D H M 1 3 " D 1 1 M 3 3 •= 0 -As * 2 i l J L 3 3 * D 1 1 M 1 3 * D n M 3 3 ' ° (uf- + A n - ) L 1 3 • A 1 3 L 3 3 • 2 1 ^ 3 * 0 M 3 3 * 0 A 1 3 L 1 3 • ( u F h A 1 ] L ) L 3 3 t 0 M 1 3 -»2itA,2M33 = 0 •.2iu , 2 L 1 3 * 0 L 3 3 * ( c o 2 + D n ) M l 3 + D n M 3 3 - 0 0 L 1 3 - 2 i * i L 3 3 • ( t o 2 t D 1 1 ) M 3 3 = 0 The determinant of the c o e f f i c i e n t s is zero, hence there e x i s t solutions d i f f e r e n t from zero. Let L ^ be ar b i t r a r y . Solving by determinants and l e t t i n g 11. ^ ^2cd»rc^(A 1 3tD i : L) * (A 1 1-»A 1 3)D 1 1 |« 2v? 2r 4 . . 2 2 . 2, J33 - ur(Axl • D n ) * (A-LT^ • A 1 3 ) D 1 T | L 1 3 L13 2«*f-4Dni\D 11 ^11 -2to2U^ - 4D1J D X 1 M 1 3 -M - - L i 3 ; l p ( 4 A 1 3 4- D u ) *. 2o 2(A 1 1D 1 ])+ ( A n 2 - A 1 3 2 ) D u 2 u»(ci - 2 D 1 1 ) D 1 1 J L 1 3 Ll3 <2L 13 33 - - i f - 3 t i • L A4(D 1 1-2A 1 1) + u 2(A 1 1 2-Aa^E^ 12A, |D 1 1) * V A U 8 ' A 1 3 2 ) ] L 1 3 = - 4 - -2iL Sim i l a r l y for 13 L14 ar b i t r a r y , M14 = -2L 1 4, M 3 4 = 2 i L 1 4 , L34 "= ' L14* L15 II 9 t. M 1 5 , % 5 « ^ s ^ i s * L35 "= *35 L15* L16 II i M16 = - m15^ L16» - m 3 # L 1 5 » L36 L17 II * M 1 ? = mi7^L 1 7, M 3 7 •= m37^ L17 • L37 - e37 L17 L18 II » M18 = -m 1 7^L 1 8, M38 = -m3?WL18, L l 8 = e37 L18* where m15 _ D l l A 2 - ~2«>f{(> + 3u,2 - 4D 1 1)D 1 1 j p 2 * /* (2<«2 • c J : f p 3 -^£(2<o2 - 3 D n ) 4^>(-3u? 4 2o7Dnl - 4 D i f ) 2 L 4D ) * 9u? - 1 2 A , 11 2,„.2 _ x . „,_„4 - 2_ ._ 2. - D u(9 « i - 12«oD i ; L) ] 12. 2*0| -35 - g ± f o 4 f > ( ^ • D n ) 4( 3 u , - 4 D 1 1 ) D 1 1 A 3 - - 2^V( V + 3u? - 4 D 1 1 ) D 1 1 m17 _ D l i r V + V ( 2 t > 2 * 4 D l l ) • 9 v " I S C A ^ T J 3 o 2 4 m, 37 - 1 / 2 ( 2 w - 3D.,.,) + y ( - 3 c i • 2 w2 D n - 4 D ! ! 2 ) 11' ] e 3 7 - i i e V V + V ( ^ * D 1 ; L) • 3 « J i D i - 4 D 1 1 Equation (8) gives i ^ e 1 * * • H 1 2 i 2 w 2 e 1 , w t 4 N 1 3(Te^ * N ^ e -J&t - E 1 3 ( U 3 1 e 1 U > t 4 Ngge1*-* X^J* 4 u ^ f * ) = 0 N 3 1 i i > 2 e i U , t • N 3 2 i l , 2 e - i u ' t 4 HsgTe^t 4 N 3 f ^ * l14t 13' 33 N a r b i t r a r y , ^12 a r D : ' - t r a r y * ^ i 3 a ^ ^ ^ t r a r y » N a r b i t r a r y , N3 l = - » u H3 2 = - H 1 2 If 33 - N 13 E 13 N34 , ( < r- E11^14 a N14 E 13 The solutions of (7) and (8) now become 13. x 3 | I 1 2 + 0 - ( L 1 3 e i u > t * L 1 4 e i t o t ) + e 3 5 l L 1 5 e ^ + L 1 6 i ? ? t ) 4 e 3 7 ( L 1 7 ? t # L 1 8 e 5 5 t ) y 1 = M ^ M ^ t ^ i U ^ e 1 ^ ^ Z = N e ^ N ^ e ^ n e^V :fn 1 11 Xf-' 13 14 z 3.-N ue --H12e •S^W^e**. — ( 1 1 ) . The Asymptotic Solutions: According to Poincare' s ^ d e f i n i t i o n , the asymptotic solutions of a set of d i f f e r e n t i a l equations are those in °k t which each term i s of the form e P(t) where * is a constant having i t s rea l part d i f f e r e n t from zero, and P(t) i s a constant or a periodic function of t. Such solutions approach zero as t approaches +aO or -<0 according as the re a l part of of i s negative or positive respectively.. From the solutions (10) of the equation of v a r i a t i o n i t w i l l he seen that the asymp-t o t i c solutions w i l l be expansible in powers of such of the +Jpt rtfvt ifirt exponentials e 1 ,.e , e which have r e a l exponents that are not purely imaginary. It w i l l be seen that not a l l the exponents are imaginary. The Numerical and I l l u s t r a t i v e Example: To construct the numerical solution of (6) l e t the unit of mass be chosen so that k 2 * r k 2 -z 1 and mo = 1. Then 2 1 . ^ "I c m= .00014 and e = 28 x 10 . Then 32 ^9 ' B = 1568 x 10 C ^  196 x 10 <->2 * 84 x 1 0 3 7 A = A - 280 x 1 0 3 7 11 33 14. A 1 3 A3]_ = 28 x 1 0 3 7 37 D l l = D13 ^ D 3 1 - D33 « E31 = E13 - x 10 3? E l l ^ =33* " 9 8 X 1 0 and i'(2) -(315 fx, 2- 2 ~ 2 \ 2 2 l ~ " ! l _ - ! i V 4 2 ( X l X 2 " i y 2 " Z l Z 2 V 2 1 ( X 3 " y3 - Z 3 ^ -2- 2 xlO 37 (2) (2) (2) 21/x 1 2-y 1 -z 2 - y 1 y 2 - z i M - a i s K 2 - y 3 1 / 2 2 xlO 37 ^ [ - 2 1 x 1 y 1 + 2 1 ( x 1 y 3 + x 3 y 1 ) + 3 1 5 x 3 7 3 " ] x 10* ( 2 ) n R^ « ^ -315x- Lz^-21(x^Z3+X3Z^)421 X3Z3 ( 2 ) R, = 1 - 2 1 x- Lz 1 + 2 1(xT tZ3tX3Z x) + 3 1 5 X 3 Z 3 N37 , 37 X 10 ,~37 X 1  X 1 0 3 7 ,~37 X 10 The factor 10 may he dropped for the subsequent compu-tat i o n and introduced again in the f i n a l r e s u l t . Thus taking J * 84, A 1 ; L - 280, A 1 S - 28, n = -14, B i n = -98, L13 Then 1 1 n/= - 1 2 5 11 P - 6 9 (H: 112 and the roots of the c h a r a c t e r i s t i c equations^ and (9) become >^0, 0, 1 ij84, ^ \ f 6 9 , •£ i>[l25 ^=+i^84, •+ i | T l 2 respectively. Only the exponents containing ± ^"p" are r e a l , hence we need determine only the constants m-j^, n^g* and © 3 5 . The computa-tion shows that m i 5. -.18, m 3 5 -.50, e , c c : 1.00 35 The d i f f e r e n t i a l equations become 15. 2 2 2< d 2x 1-2[84dy 1 280x 1+28x 34315/x 1 -y1 -7, 2 U^lx^-y^-z^ d t 2 dt \ 2 2 / \ 2 2 / 2 2 2 - 2 i [ x 3 -y 3 - z 5 2 2 d 2x 3-2|84dy 3 _ 28x 1*280x 3t2l/xf -y 1 2-z 1 21 - 4 2 y'x 1x 3-y 1y 3-z 1z 3 '"2 2/ 2 2 2 d t 2 dt \ 2 2 2 ) -315^x3 -y^ -z 2 2 S 2 v t2^84dx i _ -14y i-14y 3-315x ly i-21(x 1y 34x 3y i)+21x 3y 3 _ dt 2"^ dt k 2y 342[84dx 3 _ -14y 1-14y 3-21x 1y 1+21(x 1y 3*x 3y 1)4315x 3y3 dt*0 dt o d z _ -98Z]_-14z3-315x- LZTt-21(x1z34X3ZT t)+21x3Z3 d t ^ - ' d 2 z 3 _ -14z 1-98z3-21x 1z 1t21(x 1z 3tx 3z 1)+315x 3z 3. (12). d t 2 Let f x j- /— -i egt (n) n ( i 3 ) i y i ^ 2: Y J / V z. -- f ' Z i l n ) e n ( i « l , 3) functions in t . Substitute (13) in (12). In) "d 2x 1 ^ ^2 :{&4dy^ n * -280x^n ^  -28XL£ d t 2 dt . n 3]+2l/x 3-y -z , ' - s i 5 M n ) ^ > 1 - ! i : f ) - 4 f i n ) 4 n ) ^ - ^ ;• \ S 2 / V 2 2~/ -y \8"'4 16. d x 3 - 2/84dy^ - 28x]_ - 280x 3 8t2 dt € + • 2 1 ( X 1 ' £ f i j - 4 2 ( X l X 3 * M i " f l f a j - 315|x 3-Y3-Z3 \ U 3 1 5 X! yi + 2 1 (V 3 * ' V l ) " 2 1V 3 k=° t j ^ l x . ^ - 21(x ]y 3 4 x ^ ) - 315x 3y / £ - 0 + 1315x2,^ f 211x^3 4 Xgzy ) - 2 i x 3 z 3 £ = 0 °° V„2 , „ I~J « 0 i ~ J 7 2 1 ^ z W - 21 (x^T » ^ V" 5) - SlS^feTo — (14) • I l 1 3 3 I On equating to zero the c o e f f i c i e n t s of the various powers of £ in (14) we determine the a r b i t r a r y constants in order that the solutions w i l l have the required asymptotic form and-will s a t i s f y certain i n i t i a l conditions. The solution for the orbits which approach the equilibrium points as t approaches••+ «0 are f i r s t obtained. Equate to zero the c o e f f i c i e n t of £ i n (14). We obtain d i f f e r e n t i a l equations which are the same as (7) and (8) with x^, y^, and z x replaced by x j 1 ^ , yj^» and z } 1 ^ respec-t i v e l y . Thus we. have d 2 x ^ ^ J l H i d y J 1 ) - 280xi 1^ - 28x[1^ 0 S71 7? 1 3 17. d ^ 1 ^ - 2/84dy31^ -SSxj 1^ - 280X3 1^= 0 ""dt2" i t " d 2 y | l ) t 2|8Tdx^l ) +*4 x[ l ) + 1 4 y 3 1 ) - 0 dt^ Z-2 3 <-> o d t 2 d V ^ t 98z. (l ) 4l4z< l } = 0 — 4 — 1 * • -dt-2 u ) d ) ( i ) d z- lV • 14z, + 98z = 0 3 1 3 d t 2 The general solutions of these equations are the same as (11) with L 1 6 replaced hy I^g 1^. Those which s a t i s f y the condition for asymptotic solutions are x l - L16 e " L i 6 e 4 1 ] • • z { 1 ) - ,U>-Z l r Z3 L ^ l ^ i s an ar b i t r a r y constant and may he determined hy imposing the i n i t i a l condition x (0)= Z . Then (1) do x x (0) = 1 x{ ; - 0 for k ^  2, 3, 4, and The solutions s a t i s f y i n g the asymptotic and i n i t i a l conditions become 18. x(D, i m , y ' 1 ' ^ 1 . 4 9 e ^ x j 1 ' ^ , y l l i = 4.15 411 = ,ID , 0. 2 On equating the c o e f f i c i e n t s of £ to zero we obtain d 2 x ^ - 2|84dy^ - 280x° - 28x3*J d t 2 d t " ^ 315^x 1 "|1 "|JL j * 4 2 ^ * 3 - Z £ ^ -21 /x 3 - £ 3 ^ 3 J 2 l*J , — - {»> (>J d x, - 2484dy, -28x, - 280x, £ 3 1 3 a t 2 dt \ "2" "^J V " 2 ~ — 7 V ~ ™™2 / d 2y^° t 2>[84dx:[,J +• 14y^ t 14y™ a t 2 ~dt (.0 l>l , fit tu (0 (•) •. CO (>i ~ - S l S x - ^ - 21 ( x ^ + x 3 y 1 (> t 21x 3y 3 d2y3"J * 2j84dx£l + 14yC1° * 1 4 x 3 ^ d t 2 " dt 5 = -21x1V1 4 2 1 ( x v * x 3 y L ) + 315x^3 On substituting for X T ^ j - y f 1 ) , and z j 1 ^ in the right side we obtain . _ . . . -2^69t .2 (2) d x^ ^JeldyJ" - 280xl1lJ dt 2 dt 2 (*> d x-z -2f84dy 3 J - 28x U J 1 dt/* dt" .2 tl> d yi 1— ty *2J84dx 1 to 4 l 4 y 1 dt 2 dt 2 (0 d y 3 *2r84dx3 t*-> 4 1 4 7 l dt 2 ~dt 3 34.44e -280X3°* 2489.SQe" 2^* + 1 4 y 3 W - -50oi64e- 2 ,T69 t 1 4 y ^ \ 1 3 94.40e- 2 r a t 19. The complementary function which has the asymptotic form i s determined as in the previous case. Then (2) -f69t 42) x(2) .(2) L i e ' e x i .lat&LM I'**** - .18i69x|2) ~ z (2) 0 (2) (2) From the i n i t i a l condition = 0 y L i 6 ~ ®. T o determine the p a r t i c u l a r integral write equation (15) i n the form (>,2-280)x1- 28 x 3- 2{84>y1- 0 y 3 - 34.44e -28xi' t ( X-280)x 3- 0 y x - 2>p}4*y3 - 2489.59e- ^ 1 IX) , 2 , i») to -2\T69t 2T84>x1- 0 x3-^(X-14)y1-»- 14 y 3 « -500.64e x 1 + 2^847vx3 + 14 y^(y?-14)7% - 1394.40e- 2^ 9 t Then a* -2j69t 34.44e -28 -2>T84>. 0 2 4 8 9 . 5 9 e - 2 ^ (*2-280) 0 -2^*84^ -500.64e"* 2^ t 0 (>2-14) 14 1394.40e"*2^6_9t 2f84 > 14 (^-14) (>,2-280) -28 -2{8T7\ 0 ,- ~ 2 8 (7?-280) 0 -2J8T>v 2(8T> 0 C j ? r l 4 ) 14 0 2^84"V 14 The denominator is X \ > X ( 1 J 4 P ) •» X ( - 3 9 2 0 ) - 7 2 4 4 1 6 1 To solve put X ~ -2$69 Then x l 2 ) ^ 2 . 4 8 e ~ 2 ^ t 20. Simila.rly 4 ;E) = 3.49e -2f69t t2> - ,46e -2^69t yj s 2 ) - 8.44e -2J69t If we consider only the f i r s t and second degree terms we f i n d the asymptotic solutions s a t i s f y i n g the i n i t i a l condition x(0) = £ to he x ^ e-- r a t«r * 2.48e - 2 l ^ " t e i L x 3 - e - ^ c f * S ^ Q e - S ^ e " y 3 r : 4 . 1 5 e - , J S § t C + 8 . 4 4 e - 2 ^ t C On substituting in £ x - -1 + x.^ / 2 - x 2, / 3 - 1 + x 3, ^ i - y"i» and ^ i •= z i we obtain the parametric equations for the orbits which approach the equilibrium points asymptotically. They l i e i n the \"l - plane. The factor 1 0 3 7 which we may now introduce appears only i n the exponent. Thus f± , -1 • e--J6Tx 1 0 3 7 t e + 2 . 4 ^ ; ^ ^ ^ ^ f 3 ' U e - ^ ^ l O ^ t g " t 3.49e-2>i69 x l 0 3 7 t c>-/ Y i - 1.496-*^" c 1 0 5 ? t g t .46e- 2^9~x~To37 t C V " f 3 = 4.15e- , ) 6 9 x 1 G 3 7 t e 4 8.44e-2v69 x 1037 t £ *-L e t /s£ 1_ and t' = 1 0 - 2 0 t * " 10 Then - -1 * . l e " - 2 6 * ' t .025e-- 5 2 t' f 3 ^ 1 4 . l e - - 2 6 t ' * .035e-' 5 2 t' •= .149e" , 2 6 t' * .005e-' 5 2 t' , .415e-- 2 6 t' * .084e"' 5 2 t : ? 21. On eliminating the parameter t 1 between £ , ^  \ a n < * ^ 3' ^ 3 we obtain the general equations of the parabolas along which the electrons move to., their equilibrium positions. They are \ 2 - 10 + 25*^ 2 - 1 7 ^ * 3 ^ - 18 •= 0 f52 -.833^3+. 174^ 3 2 - 2.36^ -t . 92"?3 * 1.36- 0. The equation of the path along.which the nucleus moves is obtained from the centre of gravity equation m l f l + m2^2 * * 0 ffiA * ra2^2 * m3^3 " 0 • by substituting for £ , fz'^l' a n d /*f3» Then iz~ " M f i * K) - -.GOO^^e"' 2 6*' t .06e~' 5 2 t' ) 1 * 3 "1%- -m Wj. *^3) - -.00014( .564e" , 2 6 t'+ .089e' • 5 2 t ' ). On eliminating the parameter t* the equations become - 1.5^ 2) 2 - .QOOAJ t .00012^ - 0. The li n e a r terms, are very small, hence In order te plot these equations we must reduce them to simple forms. £ x 2 - l o f ^ + 2 5 ^ 2 - 17/ x * 3f t - 18 = 0 can be re'duc'ed to the form Y-^  = .62X by a rotation Qf axes through 1^^295' and a translation of the origin°point (-1 J;43,-. 12 ). f 3 2 - .833^2 j .174/732 - 2 . 3 6 4 + .92^ 3 • 1.36- 0 2 can be reduced to Y 3 — *05X^ by a rotation of axes through 67°32' and. a tra n s l a t i o n of the o r i g i n to the point (-.2,-1.08). 22. To construct the asymptotic solutions as t approaches - oO we need only replace - hy + \J~^> . The figure may he obtained from the above by r e f l e c t i n g about the ^ axis. The equations for this case are -1 , . l e * - 2 6 * ' t .025e*- s 2 t' -.149e« 2 6 t' - .005e> 5 2 t' - 1 i . l e - 2 6 t ' + .035e- 5 2 t' "7 3 = -,415e .26t» - .084e .52t« and the figure i s 23. 1 The three bodies move along their respective curves so that they always s a t i s f y the centre of gravity equation. 24. References: W. E. Cox: "Lagrange's Quintic for the Seutral Helium Atom," Am. Math. Monthly, Vol.XL, 1933, p. 406. (2) H. E. Buchanan: "Small O s c i l l a t i o n s of the Neutral Helium Atom near the Straight Line Positions," Am. Math. Monthly, Vol. XL, 1933, p. 532. ( 3) A, Rayl: "The Straight Line Solution in the Generalized Helium Atom," (in press). ^ D. Buchanan: "Asymptotic O s c i l l a t i o n s within the Helium Atom," (in press, &§yal Society of Canada). (5) H. Poincare:. "Les Methodes Nouvelles de l a Mecanique Ce'leste," Tome 1, Chap. VII. 

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