10 PROCEEDINGS OF THE AMERICAN ACADEMY. 



We see thus that if M(v) be the adjoint of L(u), then L(u) is the adjoint 

 of M(v). 



Analogous to Lagrange's Identity for ordinary differential expressions 

 we have here too an identity to which we may likewise give that name, 

 holding for any two functions u, v. 



an 



Lagrange's Identity. vL(u) — uM(v) = 2 TT' 



^-(•S-SM-^S) 



uv. 



This we readily verify by direct calculation. This identity furnishes, 

 as for ordinary differential expressions, a simple proof of the sufficiency 

 of the condition M(v) = for v being a multiplier of L(u). Further- 

 more we have, here as there, the proposition : 



Proposition 2. If between any two differential expressions of the 

 second order, L(u) and N(v), we have an identity of the form of 

 Lagrange's Identity, 



vL(u)-uN(v) = ^ d ~^, 



the T's being bilinear expressions in u, v, and their first derivatives, 

 then N(v) is the adjoint of L(u). 



For we get with the help of Lagrange's Identity, 



d(Si - Ti) 



u[N(v) - M(v)] = 2 



dxi 



so that u is a multiplier of the differential expression N{v) — M(v), 

 and therefore satisfies the differential equation 



Adjoint of [N(v) - M(v)] = 0. 



But u is any function whatever. Therefore the adjoint of 

 N(v) — M(v), and so N(v) — M(v) itself, is identically zero. 



Integration of Lagrange's Identity supplies, as noted for ordinary 

 differential expressions, a Green's Theorem for the expression L(u). 



Necessary and sufficient conditions that L{u) should be self-adjoint 

 are 



a< = 2 a - : » i= 1, . . . m. (10) 



For these are, by (8), the conditions that b { should equal a if and from 

 them follows b = a. For the cases, so common in mathematical phys- 

 ics, where the coefficients of the second derivatives in L(u) are con- 



