GEOMETRIC INVESTIGATIONS ON DYNAMICS. 



309 



(59) 





equations (33) and (58) may be written 



(60) Clrpl + C2rP2 + CZrPZ + + Cnrpn = 0, (r = 1, 2,. . ., 71 — l); 



(61) 



dFj dFi 

 u \dxi dxi 



Cik Cik 

 Cir Cir 



= 0, (r,k= l,2,....,n-l). 



From the (n — 1) homogeneous equations (60) we see that the p's 

 are proportional to the determinants of the matrix 



(62) 



Cn 



Cii ...Cn 



Cml . • ■ Cnl 



Cl, n— 1- ■ -Ci, n— 1- • • C I, n— 1- • • ^m; n— 1- • -Cny n— 1 



SO that, e. g., if we omit the column headed by Cmi we get 



(63) p„.:(-ir^' 



Cn . . .Cm-l, 2 Cm + 1, 1 ■ . .Cnl 



Cn ■ . •<"m-l, 2 Cm + 1, 2 ■ ■ •Cn2 



Clf n— 1- • -^771—1, n—1 Cm + lt n-l- ■ -Cm n-1 



Consider two arrays, the minors of the 2d order and the correspond- 

 ing co-minors of the {n — ^) d order in qm, where qm is the determinant 

 occurring in the above expression for pm., so that 



(64) 



Pm= (-ir+^q„ 



If we designate by Cik,ir the minor of 2d order, 



(65) 



Cik, I r 



Cik Cik 

 Cir Cir 



