486 MB. A. R. FORSYTH ON INVARIANTS, COVARIANTS, AND QUOTIENT- 



r p is less than nl/pl n p\ ; 8 M , B',,, . . . , 6 ( V M , for values 3, 4, . . . , n of /*, where 

 there is no restriction on the value of s^. Such a concomitant <f> satisfies a form- 

 equation given by (xxvii.) ; the equation (xxvi.) is satisfied when the form of <f> is 

 known, the index alone being therefrom determined. 



Now, in the form-equation the number of partial differential coefficients of $ 

 (including dfydu, . . ., 9</3v p , . . . , 9</96 M , . . . , which do not explicitly occur, but 

 which may be considered as present with zero algebraical coefficients) is 



r 4- 1 from the terms implying differentiation dependent on u, 

 r p + 1 . v p , 



Hence the total number of partial differential coefficients of <f> is 



+ 1 + "T 1 (TV + 1) + Tfo + 1); 



p=2 



all the partial differential coefficients in the form-equation, with regard to quantities 

 other than those supposed to occur in <f>, vanish. 



In order to obtain the most general solution possible as a value of < involving the 

 quantities which occur, we form, according to the usual rule, the necessary subsidiary 

 equations by means of fractions involving differentials ; the number of these fractions, 

 excluding the fraction d<f>/0, is the same as the number of partial differential 

 coefficients of <4 in the linear equation, and, therefore, the number of independent 

 subsidiary equations, being one less than the number of fractions, is 



N = r + "T'^-f 1) + T(<v + 1) 



p=Z M= 



p n 1 fi = n 



= r -f 2n 4 -}- 2 r p + 2 S M . 



p = 2 fi = 



To construct the general function <f> we therefore require N independent integrals of 

 these subsidiary equations. 



Now of invariantive functions, which have the properties of being independent of 

 one another and of involving in the aggregate all the specified quantities and 

 individually at least one of the quantities, and from each of which, on account of 

 these properties, independent integrals of the present subsidiary equations can be 

 constructed after the manner of 131 and 134, we have the following : 



