IRWIN. — INVARIANTS OF LINEAR DIFFERENTIAL EXPRESSIONS. 59 



The first part of this expression will be seen to be equal to Li, as defined 

 by (42), page 52. Since, further, 



and our expression, (43), page 53, is identical with (49), and is there- 

 fore an absolute covariant. But this is wha we set out to prove. 

 In the case of wo independent variables, m = 2, our covariant is 



( 



a f r - 7te) (da% ~ dy8x) - 



Here the second factor is itself a covariant of weight one ; so that, in 

 this case, the condition of proposition 24, page 53, would be the van- 

 ishing of — - — - — , which is not only an absolute invariant, for 



6 by dx J 



change of dependent variable, of the differential equation, but an in- 

 variant, of weight minus one, for change of independent variables 

 as well. 



I collect here for reference the covariants that we have come across 

 in the course of our work above, adding a couple of invariants from 

 Cotton's paper. 9 



^7 dxi \^A dxj 





d 2 u ^C du (daij 1 dA 



di 



i,i 



dxidxj ij dx z \dxj 2 A %] dxj 



2d} r— > and z, Udxi are absolute covariants : di and /{ are 

 dxi *r* 



defined by (45) and (47). 



9 For bibliography, see the note, page 239, of Cotton's article. The inva- 

 riant, for m = 2, -— - — -~ is also given, in explicit form, by Rivereau in 



the Bull, de la Soc. Math, de France, 29, 7 (1901); it is identical, as is easily 

 shown, with what Rivereau calls 21. 



