28 



PROCEEDINGS OF THE AMERICAN ACADEMY. 



We see, from (18), that to have A n — 1 = 0, 6 must satisfy the equation 



na n 6' + a n -\6 = 0, 



or 



n a n 



(21) 



The other coefficients are given by the formula 



(n-I)\ 



A n -k — 2< ( n - 



fg in -k)\(k-l)\ 



6V*-»On-l. 



(22) 



Substituting herein the values of the derivatives of 6 obtained from 

 (21) by differentiation, we find that An— k is times a rational func- 

 tion of the as and their derivatives, say A n —k = In—k(a) 0. Here 

 we use the letter I because the expressions in question are, in fact, 

 invariants. For let L{u) go over under u = $-U\ into Xi(wi) with 



coefficients a. Then Li(u{) will go over by u\ = - rj into A(t;) above. 



Since this is a canonical form for i 1 (w 1 ), as well as for L(u), we shall 

 have 



A n — h = In—k( a )-,- 



Comparing this with A n —k = ln—k(a)0, we get 



In—k( a ) = $I n -k{a)- 



The expressions In— k = An—k/0, k = 0, 2, 3, . . . n, are then rational 

 invariants, of the first degree, of the differential expression. Moreover, 

 they are invariants of the differential equation. 



For it will be seen from (21) that 6'/ 6 is the same for <f>L(u) as for 

 L(u) itself, and the same, it is clear, will be true of 6^ k ~ l ^/0. We see, 

 then, from (22), that I n —k, that is A n —k/6, formed for cf>L(u) is <£ times 

 I n —k formed for L(u) ; or In—k is an invariant for multiplication by </>. 



Now, further, suppose that two differential equations, L(u) = 

 and Li(ui) =0, have these invariants proportional; that is to say, 

 if L(u), L\(ui) go over by u = • ??, ui = Q\-r] into canonical forms with 



coefficients A and A respectively, then A n —k/0 = p(x)A n —k/9\- If 



Q 



now we multiply the former of these canonical forms by — , it goes 



pV 



over into the latter. We have thus the proposition : 



