﻿1092 



Mr. R. Hargreaves on Atomic Syste 



ms 



an equal angular velocity determined by the conservation of 

 angular momentum. The radial components then furnish 

 equations 



m 1 e- 2 r 1 2 (n 2 r 1 — r 1 )=/i(X)— \c n , ) ,^ 



m 2 e-V(OV 2 -r 2 )=/ 2 (X) — \c n ; ) 



where to secure symmetry the notation of (6) is altered, viz. 



/,(*)=/(»). /i(*)=/(;). 



With x and 1 for original radii, p x and p 2 for displacements 

 r 1 = ( ^-f^> 1 and r 2 = l + p 2 make X — x = p x ■+ xp 2 ; and the 

 equation of angular momentum is 



^{m^ 2 -h m 2 r 2 2 ) — (o{m. x x 2 + ??? 2 ), 



or n-a)(l-2^-Vi-2^- 2 /9 2 ). 



Tf then p 1 = —c/co 2 p u the left-hand members of (24) are 



N M {i + (9 2 -i>-Vi- 4 ^~V 2 } 



and yLtN^ " 3 {l-j-(^ 2 + 3) / 2 — 4A'" 1 /)! — 4yLtA'" 2 /9 2 }. 



Further, we have 



./I (X) - ic,=/i {a) - ic M + Ox - ^ 2 )/V (a?) 

 = N ft + ( / Q 1 -a; i o 2 )/ 1 , (a7); 

 and the equations then stand 

 (px-xpdn'^ ^- 1 N fX {(^-l) Pl -4^- 1 /)2 }, 1 



^. (25) 



(n-^ 2 l/ 2 '=^-X{( 9 2 + 3) i o 2 -4^-V 1 --4^- 2 p 2 }; 

 whereupon 



p^M 2 - i) 2 + ^%'(q 2 -l)-Apix% , = . 



represents to the first order in p, the result of elimination. 

 The small root is q 2 — l = 4yLw;~ 2 , and the ratio of displace- 

 ments associated with it is p 1 = xp 2 to the first order. The 

 large root is q 2 — 1= — x^f 2 (x)l puN^, and the ratio of dis- 

 placements /a/02/i'(#) = #Vi/2'(#)* Here p 1 and /j,p 2 are on 

 the same scale of magnitude; and, indeed, when x — 1 is 

 small, i. e. n not small, // and f 2 are opposite in sign 

 and nearly equal in magnitude, making p^-\-pp 2 = or 

 m 1 p l + m 2 p 2 — a first approximation. The sign oi f 2 (x) 

 or fix) is negative and the oscillations real ; a limiting- 

 value q = 16'12n when n is great, is found by using the 

 value of /'(l) given above (§ 12, footnote). 



This root appears in more general equations as attaching 



