52 



PROCEEDINGS OF THE AMERICAN ACADEMY. 



To translate into terms of differential expressions we must substitute 



an— k 

 a n 



in the ©'s for the coefficient p n — k of the differential equation 



«<») + pn-iW^-D + 



. =0. 



The expressions so obtained are evidently, like the ®'s, invariants, 

 both for change of dependent and independent variable, of the differ- 

 ential equation L{u) = 0. 



Next let L(u) be a partial differential expression and of the second 

 order. <j>-L(u) is to be self-adjoint. Necessary and sufficient condi- 

 tions thereto are, by (10), page 10, 



<Mi = 2 



or 



If 



2 °a 



dlog<£ 

 Ox, 



ai 



-2 



an 



dxj ' 



dajj 

 dxi' 



i = 1, 2, 



A = 



aii 



ami 



a r , 



is not zero, we may solve these equations, and get 



dlog</> _ 



dxi 



m. 



= Li, 



let us say. Necessary and sufficient conditions that these equations 

 should have a solution, log <f>, are 



dxj dxi 



i = 1 9 



& J., £i f . 



m. 



The expressions on the left are absolute invariants, for change of de- 

 pendent variable, of the equation L(u) = 0. For if we refer to (24) and 

 (27), pages 31-32, we shall find that Li = Xi — ki; so that 



