IRWIN. — INVARIANTS OF LINEAR DIFFERENTIAL EXPRESSIONS. 25 



pendent variable; an invariant for change of dependent variable is 

 an invariant of the adjoint for multiplication by <f>. 



We prove the first part of the proposition. Let 7(a) be an invariant 

 of L(u) for multiplication by <f>; and let 1(a), expressed in terms of 

 the 6's, be J(b) ; 1(a) = J (6). Let M(v) go over under v = cf> • v linto 

 M\(v\) with coefficients /3. Then by proposition 6, page 19, <f>L(u) 

 and M\(v\) are mutually adjoint. Therefore 1(a) = «7(/3). But* 



1(a) = <j>nl(a) = <f>nJ(b). 



Therefore J(P) = <pJ(b). Q.E.D. 



Expressions 7(a) that are invariant not only for change of dependent 

 variable but also for multiplication of L(u) by $ it will be natural to 

 speak of as invariants of the differential equation L(u) = 0. 



Now let 7(a) = J(b) be any such invariant. By the proposition just 

 proved J(b) is also an invariant of the differential equation M(v) == 0. 

 Therefore J (a) is an invariant of L(u) = 0; and, the relation between 

 L(u) and M(v) being reciprocal, J (a) =7(6). 



Proposition 11. If 1(a) = J (b) be an invariant of a differential 

 equation, then so also is J (a) =7(6). We shall call either of two such 

 invariants the adjoint of the other. 



It is evident that proposition 5, page 19, may be extended to in- 

 variants of a differential equation: if not itself isobaric, such an in- 

 variant is nothing more than the sum of invariants which are. 



As to a complete system of invariants of a differential equation see 

 below, page 29. 



§ 8. . Invariants of the First and Second Degree of Differential 

 Expressions and Equations. 



A problem of interest with regard to the invariants of a differential 

 expression or those of a differential equation is that of determining all 

 the invariants of a given degree. The results which I have been able 

 to obtain concern invariants of the first and second degree. 



The methods I have employed are as follows. In the first place, as 

 we have seen, we need merely consider invariants isobaric with respect 

 to each independent variable. Next, in the case of invariants of a differ- 

 ential expression, we may confine ourselves to such as are homogeneous 

 in each b and its derivatives. For if we call the coefficients of yjrM(v) 

 6's, we have 



7(6) = ^7(6). 



