16 PROCEEDINGS OF THE AMERICAN ACADEMY. 



Treating the second term on the right in the same way, and so on as 

 long as we can, we get finally 



d k u d ( d k ~h 



dxPdy< 



" dx \ Va dxv- l dyi) dx \ dx dxP- 2 dyi J 

 dx \ dxP" 1 ~dy~ q ) ^ dy 



d fdP- x {va) d^u\ d fdP(va) dQ-hi\ 



: V dxP dyQ- 1 ) + 



«-** *(£&•) + <-* 



d k (va) 

 dxPdiji 



The last term on the right is the term of uM(y) corresponding to the 

 term of vL(u) chosen. The other terms on the right are derivatives 

 with regard to x or y of expressions bilinear in u, v and their deriva- 

 tives of order less than n. Applying the same process to all the terms 

 of vL{u), we reach the result: 



Lagrange's Identity. For any two functions u, v of x and y, 



vL{u) - uM(v) = — + — , 

 ox dy 



where 5, T are expressions bilinear in u, v and their derivatives of 

 orders up to the (n — l)st. 



In the process sketched above, there is evidently much that is ar- 

 bitrary. Thus we might equally well have written 



" fy \ a dxPdyi- 1 ) dx \ dy dxP^dy^- 1 J 



d k u 

 BxPdyi 



a choice being offered at each, or at least at many of the steps of the 



process, of what the next term to be written down shall be; the last 



d k '(va) 

 term, in any case, being evidently as above (— l) fc - — — - u. So 



that the S and T in Lagrange's Identity are far from being uniquely 

 determined. 4 



Corresponding to proposition 2, page 10, we have here also that if 

 between any two differential expressions there holds an identity of 

 the form of Lagrange's Identity, then each is the adjoint of the other. 

 This justifies the assumption made on page 14 above, that L(u) was 

 the adjoint of M (v). 



4 The process employed first above is that suggested by Darboux, Surfaces, 

 2, 73, note. His identity numbered (7) on page 72 is derived by some other 

 of the many possible processes. 



