IRWIN. — INVARIANTS OF LINEAR DIFFERENTIAL EXPRESSIONS. 29 



Proposition 14. The expressions 



J-n—k = — n > * = 0, 2, o, . . . n, 



where the ^4's are the coefficients of the canonical form into which L{u) 

 goes over by u = 6 ■ n, form a complete system of invariants of an or- 

 dinary differential equation ; a complete system, that is, in the sense of 

 equivalence, as explained on page 20. 



Next let / be any rational invariant, of degree (x, of the differential 

 expression. 



V^l \(l n , On , • • • Oln— 1, dn— 1 > • • • &n—ky Qn—ky • • •) 



= 1 \A n , A n , . . . U, 0, . • . An — kt • ' ' ■"■n — k) • • •) 



which, since J is homogeneous, is equal to 



fiy-T ( An An ' n n An ~ h An ~ k \ 



-\T'j'" ' '"' e ' " ' e ' " ' J 



= 6*1 (I n , I n i, . . . 0, 0, . . • I n — k> In—k,l> • • •)> 



if we put In—k, i — An—kl&- The expressions I n —k, I are, like I n —k, 

 rational invariants of the first degree. This we shall prove in a mo- 

 ment, and thus get the proposition : 



Proposition 15. Every rational invariant of a differential expression 

 under change of dependent variable is a rational function of the 

 rational invariants of the first degree 



j A n —k j A n —k 



* n—k — n > *■ n—k, I — f) ' 



where the A's are the coefficients of the canonical form into which L(u) 

 goes over under u = O-v, and satisfies (21). 



1 \flti, Q>n > • • • &n — 1» O'n — 1 > • • • ^n — k> • • • ^n — k> • • •) 



= ^Unt -*nl» • . . 0, 0, . . • in—ky • • • *n — k, ly • • •)• 



In particular, if / be a polynomial, it is a polynomial in these invariants 



as well. 



We note that 



. n(n — 1) n— 1 2 

 na n a n -2 H ^ (a n 'a n —i — a n a n —i') - — a n —\ 



In-2 = " • (23) 



na n 



This is the invariant of proposition 13, page 26. 



It remains to prove that In— k, I is a rational invariant of the first de- 



