IRWIN. — INVARIANTS OF LINEAR DIFFERENTIAL EXPRESSIONS. 13 



(n-k) ! _ . dP pq . f dQ p + q = n-k, 



^— —j-va pq =Pp-i >q + -r-^ + terms coming from— -, r , 



p\q\ dx dy p=l,2,...(n—k), 



va n - k = — 7T h terms coming from — , 



ox oy 



k = 1, 2, . . . (n - 1) ; 



vann = — — + a term coming from — -. 



dx dy 



Operate on each of these equations with 



Qn—k 



(-l) n -*T— — , k = 0, 1, . . . n, 



and add. On the left we get the expression 



M{v) = 



This we define as the adjoi?it of Z(?/). On the right we get zero. For 



dP 

 consider the terms coming from — . These give 



a™— ^ r> Q n _/ c -)_ i p 



in — k ?3- 



n Z^ 'IZf dn—kp . JL 7 JZf 



If, in the second sum, we put p = p' — 1 k = k' + 1, it goes over into 



the negative of the first, and the two cancel each other. Similarly 



■\f\ 



for the terms coming from —^-. A necessary condition, then, that v 



should be a multiplier of L(u), is that it should be a solution of the 

 differential equation M(v) = 0. That the condition is also sufficient, 

 as well as that P and Q in (13) are not uniquely determined when v 

 is given, follows just as for expressions of the second order. As to the 

 former point, we need merely notice that each of the P's itself occurs 

 in one only of the equations above connecting the as with the P's 

 and the coefficients of Q, in an equation containing the derivative of a 

 P the sum of whose subscripts is greater, that is of a P which may 

 be supposed to have been already determined from the preceding 

 equations. 



Writing the adjoint as 



