48 PROCEEDINGS OF THE AMERICAN ACADEMY. 



%i =: ?i\Xi, . . • x m ), % = 1, Z } . . . m ; 



multiplication by -^; 



v = J 'V\. 



Proof. 1 Make the change of variables in question in Lagrange's 

 Identity, 



vL(u) - uM(v) = 2 jr*, 



. axi 



where, as we remember, the <S's are bilinear in u, v, and their deriva- 

 tives of_orders up to the (n — l)st. Then the *S's go over into expres- 

 sions S bilinear in u, v, and their derivatives, with regard to the £'s, 

 of orders up to the (n — l)st, and we have 



dxi ~i dxi d£j 



If we divide this equation through by «/, we shall find that we may, 

 without altering the value of the right side, put everything on that side 

 under the signs of differentiation with regard to the |'s, thus getting 



L(u) M(v 





Here we have, between — j^- and — j— , an identity of the form of 



7 The proposition in the text is given by du Bois-Reymond in the article 

 Crelle, vol. 104, already referred to in the note on page 12. His proof, which is 

 based on the Green's Theorem, or integral form of Lagrange's Identity, runs 

 essentially as follows : 



J • ■ • J"[vL(u) — uM (v)]dxi . . . dx m = U, 

 U consisting of terms with less than m integrations. But 



f ■ f[vL(u) - uM(v)]dxi . . . dx m —J- ■ -f[vL(u) - uM(v)] j dfj. . .<# m , 



and so also we may transform U to, say, U. This gives us 



f...f[v^-u^']d! 1 ...d! m =U, 

 from which relation, of the form of a Green's Theorem, we infer, just as from 



a relation of the form of Lagrange's Identity, that ■ — y- and — j — are ad- 

 joint. I have preferred to base my proof on Lagrange's Identity itself. 



