6 PROCEEDINGS OF THE AMERICAN ACADEMY. 



5, a treatment here" followed, or to Wilczynski, who devotes a chapter 

 to the subject. Further, the ordinary differential expression may be 

 looked upon as a special case of the partial differential expression dis- 

 cussed below. 



Let, then, our differential expression be 



t/ x d n u fr-H . d n ~ 2 u nn m 



£ <*> =an d^ + an ~ l dx^ + an ' 2 dx^ + '" + ^ (1) 



We define as its adjoint the expression 



M(v) = 



(-1) dx» +{ } ck—i + ^ ' dx"-* 



+ . . . + a-ov. (2) 



If we write M (v) also as 



u , . , d n v , d n ~ 1 v , 



M (v) = fen 5^ + n ~ 1 ^= i + • • • + 6 ° v ' 



the 6's will be given by the following formula : 



& n _ fc = (- 1)» 2 (~ i) 1 m-^i^-zu ~d^=r- (3) 



J=0 



(n - k) 1 (& - /) ! dx k ~i 



We may establish next, for any two functions, u, v, Lagrange s Iden- 

 tity, 



JO 



vL(u) — uM(v) — -=- , 



where S is bilinear in u, v, and their first n-1 derivatives. From this 

 by integration would be obtained a Green's Theorem for the particular 

 differential expression in question. Further, if a relation of the form of 

 Lagrange's Identity, 



vL(u) — uN(v) = j— , 



exists between two expressions of the nth order, L(u) and N(v), then 

 N(v) is the adjoint of L(u). For we shall have 



u[N(v)-M(v)] = d(S ~ T \ 



and therefore N(v) = M(v). This follows from the proposition, the 

 truth of which is obvious : 



