292 



WILSON AND MOORE. 



This shows that V/is co variant of order 1. To differentiate a system 

 X of order 1 we have 



dX= V.yrf(X) + f/V.y(X) 



dx'VX= Vxy[f/x-VxyV,(X)] + (/x-v.Vxy (X) 

 VX = V:.yVxy:(vX) + v^v^y (X). 



15. Solution for the second derivatives.^^ As we are working 

 with a fundamental adjoined quadratic form I,a^^dxrdx„ we regard the 

 ttr, and their derivatives as known. We may then write 



ddr, ^ . d{aij) , V / \ 



dxt dyk 





jdXrdXt 



dxsdx 



s'J^'^t J 



and solve the six equations obtained by permuting /•, s, t for the six 

 derivatives d'^yi/dxrdxt as unknowns. We have 



^— = Siy yri^tiysk — h ^iA(lii) 



oXs dyu 



^- = ^iikitiysnrk — — + ^ij {(lij) 



oXr dyi: 





dxrdx 



d~yi 



dXtdXr 



7.; + 



dXsdxt 

 dXsdx, 



Iti 



Hence 



durt dats da,-, 

 6X3 dxr dxt 





yriitjisk + ytiysijrk — yny.jytk 



dyk 



+ -i;(«w) 



d-yi I d~yj 



To-+ - — ^7a- 



dx,dx 



dXsdXr 



But as (oij) = (an) we have, 



'ii\fliv ^ . la = ^ayP'ii) . . ya 



bxJdXr 



dXsdx, 



19 The solution for the second derivatives, though cumbersome, is exceed- 

 ingly important for it is through this substitution that the Christoffel symbols 

 actually arise (see Christoffel, Gesammelte Werke, or Crelle J. Math., 70, 46.) 

 The method followed in so many books, viz., to write down the Christoffel 

 symbols without any preliminaries seems decidedly artificial. We may point 

 out that ivhen the analysis is carried on in matrical notation, as below, the 

 elimination suggests itself much more readily than ivhen we have so many sub- 

 scripts and summation signs to manipulate as in the ordinary derivation. 



