290 WILSON AND MOORE. 



the transformation of the differentials is hnear; — thus 



dx ' dx ' 



dxi = Z~dyj = ^jCijdiji, Ca = — -* , 

 ^Vi 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 



_ dXi _ dpi 



dijj dxj 



now hold we may define co variant and contra variant systems of 

 order k as those for which 



Y - V fY \ djn^Jlh ^J[ik /r>QN 



-^ti 12 • • • ifc ~ ■";! h' ' • ik K-^ii h • • • ik) ^ ^ ' ' ' -^ ' K'^^J 



dXi^ dXi2 dx 



tjt 



(Y- ■ .\ — y.. . Y- ■ ■ ^J^^:^ . . .^:^k /r>Q/x 



\^^ II 12 ■ ■ ■ ikJ ■"n n • • • Jk '^n n • • • ]k „ „ „ ' W^ / 



dyh dVn dyik 



Z(- - • -^^^ = ^n «... ikiX^nn- ■ • /.)) pl^l. . .^^ (30) 



dyh dyi2 dyjk 



(X('-i ^2 ■ • • H^) = 2,1 yj . . . y, X^n h- --ik) ^121^111... ^lih. (30') 



OXj-^ UXj2 ^"^ik 



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

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



dXi ^ dyj 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 X,- of the first order, the 

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



