290 



WILSON AND MOORE. 



the transformation of the differentials is hnear; — thus 



dxi = 



dXi 



dyi 



dyj = ^jCiidyj , 



-t] 



dxi 



dyj 



with the difference over the algebraic theory that the coefficients c,-,- 

 are variable. As the work done to this point does not involve deri- 

 vatives of the c's or in any way depend on their constancy, the whole 

 work remains valid. As the particular relations 



"ij 



dXi 



dyi 



7 



J« 



dyi 



dXj 



now hold we may define covariant and contravariant systems of 

 order k as those for which 



-^ti 12 • • • it ~ ^n 



{Xi, 



n ■ 



t* 



Z(il i2- • • u) — -v . . 

 —'31 n • 



(X(Hi2---;A:1) = S,-,,-2 



Ik 



(A/i j2 • • • ik) 



) ~ •^n j2' • • ik -^n 



32 



Ik 



(X(" 



n 



Ik 



■ ik)) 



_J'(;'i ;2- • • ik) 



If we have a function of the variables, the derivatives /{ = df / dxi 

 form a system of the first order. We know that. 



^ = ^.^^11 

 dxi dyi dxi 



' \dXjJ dXi 



since df/dyj = (df/dxj) by definition. Hence we see that the first 

 derivatives of any function (system of order 0) form a covariant 

 system of order 1. 



If we have a general covariant system Xi of the first order, the 

 derivatives of the elements of the system with respect to the variables, 



