DERIVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS. 477 



Substituting for u, u\ ",... from (58), and expanding with a retention of terms up 

 to the first order, we have, as the additive part of <f> which is given by those terms 

 arising in connexion with t* ( " ) , 



Combining these and comparing the two sides, we find that the finite term on each 

 side is <f> ; and the remaining conditions therefore are 



Reverting to the original variables u and z, we may write these equations in the forms 



(n - 2m - 1) W 2 + 2<r< = 



(62). 



The latter of these two equations determines the form of a covariant <f> ; the former 

 determines its index <r. 



129. The process of obtaining the differential equations satisfied by the function <f> 

 of (61) is similar to the foregoing; and the result of the work is that the general 

 concomitant <j> of index X satisfies the equation 



+T S [( + ,.) e a ?* u ,l . (xxvl), 



M = ,=o|_ t "'<' J 



which may be called the index-equation, and also satisfies the equation 



p = - 1 r-nl/p! n-p! -1 



P 



=T s 



^ = 1 I = I 



which may be called the form-equation. 



The equations (62) are at once seen to be particular cases of (xxvi.) and (xxvii.) 

 for concomitants <, which involve u and its derivatives alone. For the identical 



