DERIVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS. 483 

 Let A M denote the operator 



of the form-equution, so that 



4^=0 .......... (64); 



and let V M denote the operator of the index-equation 



=0 



so that 



V M < = A< .......... (65). 



Since <f> is homogeneous of degree p in the quantities 6, we have 



When this is multiplied by /x, and subtracted from (64), then 



so that, if we denote by !! the operator 



" = ?,">' 3e<?' 



we have 



n^="<A .......... (66), 



which may be called the grade-equation of <f>. 



186. Denoting by i/> the Jacobian of 6 M and <j>, we have 



The index of ^ is p. + 1 + ^ 8 that it has to be shown that if/ satisfies the two 

 equations 



and it follows from these that 



n^ = (> + 

 3 Q 2 



