GEOMETRIC INVESTIGATIONS ON DYNAMICS. 



313 



equations (71) may finally be written 



dF\n _ Pm dF^.A _ /dFn Pn dFn-l \ 



dpm Pm-1 dpm J \dpn Pn-1 Qpn J 



BFi Pi dFm-\ 



(72) 2 



m= 1 



dUk dUrJ m=l t=l...m-l 



dpm Pm-1 dp„ 



dPmdfm 

 dUr dVk 



\dUr dUk 

 dPm df: 



dUk Sut 



= 0. 



In (72) all the coefficients in parentheses may now be taken as 

 independent quantities, so that for ^72) to vanish identically, we must 

 have the vanishing of all the expressions in brackets. Hence 



dFi dFm-l . 



Pm-\ pi —^ = U, 



dp„ 



dpr, 



i = I,. . ., m — 1, m -\- I. . .11 



m = 1 ^2 ->.. .-^7i 



dFm-l 



By successive elimination of — ^^, we may write these 



dpm 



dFi dFk ^ fi, k = 1,. . . , VI - 1, TO + 1 . . .?? 



dpm dpm \ VI — I, Z,. . . .,71 



or 



(Pk Fi - Pi Fk) = 0, 



/, k = I,. . ., VI — 1, ??? + 1- • n 



dpm " " " " V "^ = 1'2, , w 



so that (pkFi — piFk) can only contain piSLndpk, and hence, 



(73) /)A- Fi - Pi Fk = aik ipi, Pk, .ri, .T2, . . . , .T„), {i, k = 1,2,. . ., n) 



(i ± k) 



where 



(74) . (Xik = — (Xki. 



From (72) we further have 



dFn _ Pn dFn-l ^ SFm _ Pm dFm-1 / ^^ ^ J _^ 2 -> -^ w) 



dpn Pn-l dpn Bpm Pm-\ dpm 



These are conditions on the forms of the as in (73). By successive 

 subtractions, these may be written 



