26 PROCEEDINGS OF THE AMERICAN ACADEMY. 



Now consider the terms of 1(b) homogeneous of any given degree in 

 any given one of the b's and its derivatives. The corresponding terms 



of 1(b) will, since the b's are simply the b's multiplied by ty, be homo- 

 geneous of the same degree in the given b and its derivatives ; whence 

 it follows that the terms of 1(b) in question will themselves constitute 

 an invariant. 



The result just obtained enables us to determine at once all linear 

 invariants of a differential expression. For such an invariant may 

 now be taken as containing one of the b's and its derivatives only. 

 Then if we consider any of the derivatives of the highest order of that 



b occurring in 1(b), 1(b) will evidently contain uncancelled the same 

 derivative of y}r; so that, if we are to have 1(b) — xj/^IQ)), I can con- 

 tain no derivatives of b at all. (Similar considerations would show 

 that an invariant of any degree, involving one of the b's and its deriva- 

 tives only, is essentially nothing more than a power of the b.) 



Proposition 12. Essentially the only linear invariants of a dif- 

 ferential expression are the b's themselves, all others being linear com- 

 binations of these invariants. 



The general problem, apart from this simple case, may be attacked 

 by the use of Lie's methods, as illustrated in Bouton's paper in the 

 American Journal of Mathematics, vol. 21. The complete system 

 thus obtained of linear partial differential equations, whose solutions 

 are the invariants sought for, takes on, in the case of invariants of a 

 differential expression, a particularly simple form if everything is 

 expressed in terms, not of the as, but, as above, of the b's and their 

 derivatives. I bring together here the results I have obtained by the 

 use of these and such other methods as suggested themselves, in each 

 particular case, as appropriate. 



Proposition 13. Essentially the only invariants of the second de- 

 gree of an ordinary differential expression are, besides powers and 

 products of the b's themselves, those of the form 



dbi . , dbj ^ 



dx 3 l dx ' 



of a partial differential expression in two independent variables, those 

 of the form 



dbi , . dbj dbi , , dbj 



dx dx dy dy 



bi, bj being any two of the b's. 



Essentially the only invariant of the first degree of an ordinary 



