20 PROCEEDINGS OF THE AMERICAN ACADEMY. 



For make, in Lagrange's Identity, 



vL(u) - uM(v) = - + -, 



the change of variable, u = yjr-v. S, T go over into expressions S, T 

 bilinear in 77, v and their derivatives of orders up to the (n — l)st. 

 This gives us, 



, u ,, dS , df 



But the existence of an identity of this form between the two expres- 

 sions A (77) and ijrM(v) shows that they are mutually adjoint. 



The coefficients of the adjoint, the b's, are then invariants. They are 

 linear in the as and their derivatives; cf. the formulas for them (15). 

 Moreover, it may be shown that they are essentially the only linear 

 invariants (see below, page 26). In terms of these invariants and their 

 derivatives — which latter, however, are not invariants — every in- 

 variant may be expressed rationally and integrally, simply because the 

 as and their derivatives can be so expressed. 



Further, the b's form a complete system of invariants. This phrase 

 we use in the following sense. Two configurations are said to be equiva- 

 lent with regard to a given set of transformations if it is possible to find 

 a transformation of the set that takes the first over into the second, and 

 another that takes the second over into the first. A complete set of 

 absolute invariants is a set such that if two configurations have the in- 

 variants in question equal, each to each, then the two are equivalent. 

 In the case before us we have to do with relative invariants. 



Proposition 7. The linear invariants, the b's, constitute a complete 

 system of invariants ; that is to say, if the linear invariants of two dif- 

 ferential expressions are proportional, the expressions are equivalent. 



Let L{u), A (77) be the two differential expressions, M (v), Mi(v) their 

 adjoints. By hypothesis the coefficients of these latter are propor- 

 tional; that is, each coefficient of Mi(v) is, say, Q(x, y) times the 

 corresponding coefficient of M(v). Therefore M\(v) = 6-M(y). 

 Now make in L(u) the change of variable u = 6 • 77, and let it go over 

 thereby into Ai(rj). The adjoint of Ai(^) is, by proposition 6, 6 times 

 the adjoint of L{u), that is 6 • M(v), that is, Mi (v). But Mi (v) was the 

 adjoint of A(n); so that Ai(?7) and A(?/), being each the adjoint of 

 Mi(v), must be identical; L(u) then goes over, under u = 6 -v into 

 A (17). Q. E. D. 



It is of interest to inquire after processes for deriving, from given in- 



