IRWIN. — INVARIANTS OF LINEAR DIFFERENTIAL EXPRESSIONS. 39 



(see proposition 16, page 30) to an inquiry after a process analogous 

 to the process by which from an invariant I of degree one, or, more 

 generally, of degree k, of an ordinary differential expression, we de- 



n (1 1 



rived a second, V — I. The inquiry is answered by the fol- 



lowing proposition : 



Proposition 18. If I be an invariant of the kt\\ degree of a partial 

 differential expression, then so also are 



dl , T 

 Vx + kKj > 



*i, *2 being defined by (30), page 33. Further, U p + q = n — 1, then 



a/ 



dy 



dl dl k 



ap+hq di + ap ' q+l d^l~n apql 



is an invariant of degree k + 1. 



We notice that the first two of these invariants may, with the nota- 



tion of (30), be written as p— (0*1), -^ — (0*7), just as for ordi- 

 nary differential expressions the derived invariant may, with the 

 notation of (21), page 28, which corresponds to (30), be written 



Proof. The first of the invariants above, formed for the transformed 

 differential expression, is 



£ (**J) + k-K^I = ty*-l ft I + If* ^ + k ( Kl - \ ft\ pi 



ox dx dx \ \p dx J 



= #»(g + wr). 



So for the second invariant. To get the third of the above invariants, 

 multiply the first by a Pl +i, 9l , the second by a Pl , gi +i, and add. This 

 will give us, since each of these multipliers is itself an invariant — for 

 Pi + q\ = n — 1, (30) — an invariant of degree k + 1; and by (30) 

 that invariant will be the third of the expressions above. 



