EQUATIONS OF ELECTRON MOTION 



171 



There is no reason to foresee a priori that the functions Xi(t, ai, ■ - ■ , /Js), 

 • • • , TTsit, ai, • • • , /Ss) determined in this way, by means of the complete 

 solution of equation (27), satisfy the differential equations of motion (11) 

 and (12). Nevertheless, they actually do satisfy those equations, as we 

 proceed to show. 



By equations (28), we have the relations 







dt 



d'W 



+ Z 



d'w 



dan dt ' ni=i dan dx„ 

 On the other hand, by (27) and (29), we have 



(30) 



= 



dW 



da„ 



d^W , y _ 



dan dt „,=1 ^TT 



+ H(XI, X2, X3, TTl, TTi, TTs 



,o] 



dH dTTr, 



d'W 



dan dandl 



+ E 



dH d'-w 



1 diTm dan dx„ 



(31) 



The determinant 



d'W 



dai dxi 



dW 



daz 5iiCi 



d-w 



dai dxs 



d^W 

 das dXi 



is not zero. For if it were, we would have a relation of the form 



^dW/dXi, dW/dX2, dW/dXs, .Vi, .V2, .T3, /] = 0, (32) 



independent of the a's. Now equation (32) is obviously distinct from (27), 

 since it does not involve dW/dt. Hence, the vanishing of the determinant 

 would imply that the function H'(.Ti, .T2, ^3, /, ai, a^, a^) satisfies two distinct 

 partial differential equations of the first order. This, however, is impossible 

 when IF is the complete solution of (27); for an essential part of the concept 

 of the complete solution of a differential equation is that the elimination of 

 the arbitrary' constants, from the solution and the equations obtained by 

 differentiation, shall result in the given differential equation and no other. 

 It follows, therefore, from (30) and (31) that 



dTm 



Xm. 



We also have, by (29), 

 d'W 



TTn — 



dXn dt 



+ E 



d^w 



= 1 dXjndXn 



Jyfn 



d'w 



dXndt 



+ 



y d'W dH_ 

 m=l dXm dXn dlTm 



(33) 



" Since the function W{xi, X2, xa, I, ai, 0:2, as) satisfies equation (27) identically in the 

 x's, t, and the a's. This remark applies also in the case of equation (34). 



