162 



BELL SYSTEM TECHNICAL JOURNAL 



n+X2 -\-xz\ -T- 

 dxi 



( ,\(E-Vf-mlc'J'\ 



2 4il/2 



= ^[{E-Vy-mUl 

 0X2 



\ [_ dX2 oxz J |_ dxi 



Ml 



5ar2 



(17) 



dXi 



[(E-Vy-moc] 



2 4il/2 



, n , '2 1 /2,_i/2 /r5-4i 6.431 / [dAs dA2l\ 

 \\_dx3 dxij \_dX2 oxzj/ 



The equations which correspond to (17) in the Newtonian case are most 

 readily obtained by going back to the Newtonian differential equations of 

 motion and employing the integral 



mov'^/2 + V — E. 



An easy calculation, which is entirely parallel to the foregoing, gives us the 

 following system of equations: 



ni/2N 



[1 + :*;2 + X3 



3 J 1— ( ^2 

 dx\ \ 



E - V 



LI + X2 + acs J 



dxo 



(E - V) 



+ [2wo(l + X2 + X3 )] 



'2vi-l/2 



X3 



-1/2 _^ / / r E - 



dXl \'l\+ X2 



[1 + x'i'' + x's ] 



+ [2mo(l + X2' + x^')r'' ([^^ 



^£3 



. dX2 



- V 



dA, 



dX3 



-|l/2 



dAo 

 _dxi 



dAi 

 6x2 



+ Xs J 



_ djU' 

 dxi 



(170 



— X2 



dXs 



dAs 

 8x2 



{E - Vf 



dA2 



dXs 



On comparing the systems of equations (17) and (17'), we get the follow- 

 ing useful theorem. 



If the constants E, E*, mo, m*, k, and the functions {of Xi, X2, xs,) V, Ai, A2, 

 A3, V*, Ai*, A*, A* are such that we have identically 



{E - Vy -mlc = k\E* - V*), 



dA3 _ dA2 _ k 



dx2 dxz c{2mo*) 



1/2 L dX2 dX3 J 



