24 PROCEEDINGS OF THE AMERICAN ACADEMY. 



that is, by (20), since p+p+q+q=n— k, into 7 P +p, q +q- We 

 have then, S pg = 7p+p, g+g. Comparing this with the formulas 

 connecting the d's and c's, d pq = c p + Pt q +q, we see that for these 

 values of the d's Lj(u) is an expression that goes over under u == ty,j] 

 into the same function of the 7's that Lj(u) itself is of the c's; in ot 1 r 

 words, it is an absolute covariant. Inserting then these values of the 

 d's in L](u), replacing the c's by the as, and multiplying through by 



—, we get the proposition : 



Proposition 9. The expressions 



j = 0, 1, . . . n, 



are absolute co variants for u^=^r-rj. Here p, q are any given positive 

 integers (or zeros) subject to the condition : p +q = n — j. 



For j = n, we get L(u) itself. 



For ; = : a pq u, p + q = n. 



■n • -1 / & u , du\ , 



Forj=l: n\a p+ i tg — + a Pt9+ i—\ + a pq u, p + q = n-l. 



We note that these covariants are what might, in accordance with a 

 nomenclature we are about to introduce, be called covariants of the 

 differential equation. 



§ 7. Multiplication of L(u) by <f>; Invariants of a Differential 



Equation. 



Let us now consider briefly a second transformation to which a 

 differential expression may be subjected, namely, that of multiplying 

 it through by a function <f> of the independent variable or variables. 

 Represent the coefficients of <f>-L(u) by a's. Then we define as an 

 invariant of this transformation an expression 1(a), such that 

 7(a) = <£M/(a). 



Between the invariants of L(u) and those of M(v) a simple relation 

 exists. 



Proposition 10. An invariant of a differential expression for a 

 multiplication by <£ is an invariant of its adjoint for change of de- 



