8 PROCEEDINGS OF THE AMERICAN ACADEMY. 



Here we make once for all the convention aij = a ;t -. Let us inquire as 

 to the condition that a function of the x's should be a multiplier of 

 L(u), the term being defined as follows: 



Definition. By a multiplier of L(u) is meant a function, 



v(x v . . . x m ), 

 such that 



vUn) = |'g, (5) 



where the P's are linear differential expressions of the first order. 

 First suppose that v is such a multiplier. Writing 



_ ^ du 



i ' 



we see that we must have 



2va i} - = pij + pa, (6a) 



«* = 2 d -B. + » (66 > 



a 2 



Operating on the first of these equations with - — - — , on the second 

 with — - — , summing and adding to the last equation, the right 

 side cancels out and we have left 



^ d 2 (aijv) s? difliv) 



2 0\UiV) 

 TE-+"" 1 



».7 



dxidxi -r* dx, 



Our assumptions here are that the second derivatives of the ofj/s, the 

 first of the a/s, that come in question, exist, and, if we desire that 

 property in the coefficients of the equation last written, are continuous. 

 The left side of that equation is, like L(u), a linear differential expres- 

 sion of the second order; we define it to be the adjoint of L(u). 

 Definition. By the adjoint of L(u) we mean the expression 



d 2 (ajjv) -^ djajv) 

 dx{dxj ~* dxi 



We have proved, then, that a necessary condition that v should be a 



KW^S'-J^f C7) 



