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) + (/v.y(X) 

 (Zx-vx = Vxy[(Zx-Vxyv,(X)] + rfx-VxVxy (X) 



VX = VxyVxy:(VX) + VxVxY- (X). 



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

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

 ttrs and their derivatives as known. We may then write 





dxt 



drjk 





_dXrdXi 



dxgdxt 



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

 derivatives d^yi/dxrdxt as unknowns. We have 



dttrt „ d{aij) , V ^ ^ 

 = Si; yrfYti-lsk — h ^iAaij) 



dx. 

 dats 



dtjk 

 d{aij) 





dXrdXs 



dXsdx 



sV.tt 



dXr ayk 





dxtdXr 



dXsdXr 



Hence 



dart , dots dOr 



+ 



dXs dXr dxt 



— v... 



yriYuysk + ytiYsjyrk — yn^sj-y 



tk 



d(aii) 

 dijk 



+ 2,7(0,7) 



7o+ . .. Tfi 



dxrdx 



dXsdXr 



But as (ai,) = (a,,) we have, 



'ijifliV 



dXsdx, 



yti = ^iiKciij) 



dXsdXr 



yti 



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 maj^ point 

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

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

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



