12 PROCEEDINGS OF THE AMERICAN ACADEMY. 



§ 3. Partial Differential Expressions of the nth Order.% 



For the general case, partial differential expressions of the nth order, 

 we shall content ourselves with considering differential expressions in 

 two independent variables. The formulas themselves suggest what the 

 extension to the case of a greater number of variables will be, and this 

 suggestion leads throughout to the correct formulas for the latter case. 

 We emphasize once for all this remark, which applies to the whole of 

 the rest of this paper. 



We make use here of the following notation : 



k=Op=0 



pi ql pq dxPdyi ' 



q being defined by p + q = n — lc; while the subscripts of any a de- 

 note respectively the number of differentiations with regard to x, y in 

 the derivative of u to which that coefficient is attached. We may pass 

 from this notation to that employed for the second order by writing, as 

 subscripts, p ones and q twos. 



We inquire first, as for expressions of the second order, as to the ex- 

 istence of multipliers of L(w),.that is of functions, v, such that 



,, , bP , bQ , iON 



vL(u) =dx- + i> (13) 



where P, Q are linear differential expressions of the in — l)st order, 



with a similar expression for Q. If v is to be such a multiplier we must 

 have 



n ! ' , , bQ p + q = n, 



— : — : va p „ = Pp-i, q + a term coming from — t r * 



pi ql ™ ^ ' * ° by p = 1, 2, . . . n, 



vo.o n = a term coming from — ; 



2 See Darboux, Surfaces, book iv, chapter 4, and, for the second order, 

 chapter 2 of an article by du Bois-Reymond in Crelle, vol. 104 (1889). Dar- 

 boux makes use, to obtain the condition for a multiplier, of a very general 

 formula, of which we here deduce the special case we require. 



