IRWIN. — INVARIANTS OF LINEAR DIFFERENTIAL EXPRESSIONS. 55 



conditions that the coefficients of the derivatives of the (n — 3)d, 

 (n — 5) th, etc., orders in <f>-L(u) should be (— l) n times the corre- 

 sponding coefficients of its adjoint. (By proposition 4, page 15, we need 

 merely consider the orders n — k, where k is odd.) These conditions 

 would, by (15), page 14, be the vanishing of expressions bilinear in 

 the a's and their derivatives and in <f> an d its derivatives ; that is, after 



substitution for the derivatives of (/> from the equations — = L\4>, 



(J*As 



— = 1,2$, and from the equations obtained from these by differenti- 

 ation, and after division by </>, of rational functions of the a's and their 

 derivatives. And the question would suggest itself as to whether 

 these latter were invariants. 



§ 17. The Covariant 2 ( ~^ ~ ~^ ) dxiSxj. 



We return now to the proof that the expression (43), page 53, is a 

 covariant for change of independent variables. A proof of this fact 

 is to be found in a paper by E. Cotton, Sur les Invariants Dijfirentiels 

 de quelques Equations lineaires aux derivees partielles du second ordre, in 

 the Annales de l'Ecole Normale, 3e sene, vol. 17 (1900), pages 211-244. 

 Cotton's methods are based on the theory of quadratic differential 

 forms. It is perhaps worth while to obtain the result we are interested 

 in independently of that theory, as may be done with no great difficulty. 

 I shall therefore give such a proof, following in general the steps by 

 which Cotton reaches his result. I retain in part his notation. Fur- 

 ther, a dash over an expression shall indicate that it is the same function 

 of the a's, the coefficients of the transformed differential expression, 

 that the expression without the dash is of the a's. 



First, then, the expression 



17^» \VAdXj) 



is an absolute covariant. Here A stands, as usual, for the determinant 

 of the a;/s, an invariant, as we know, of weight two. The proof goes 

 as follows. Making use of the formulas (36), page 41, for the a's, we 

 get 



