Fundamental Theorem of Invariants. 187 



D 2 (another differentiant in x) will be the leading co- 

 efficient of the covariant 



and so on until we come back to the first Sturmian remainder of 

 (<£,y) 1 , the irreducible part of which (or we may call it the 

 Sturmian Auxiliary Proper) is the Hessian differentiated down 

 from being of the degree 2i — 4 to the degree i — 2, i. e. to half 

 of what it was at first ; and so in like manner every Sturmian 

 Auxiliary Proper is, so to say, a Covariant differentiated down 

 to half its original dimensions. 



The above invariant and the following covariants may be 

 called V , Y 1? Y 2 , . . . respectively. The interesting point in 

 question is that (to numerical factors pres) 



and so on. 



So more generally for any two functions f(x,y), (j>(x,y), 

 the irreducible part of the remainders obtained by common- 

 measuring them with respect to x will all be derivatives in 

 regard to x of covariants of the two given quantics. If we 

 take for our quantics 



(a, b, c, . . .A, k, ly^x, y) f : (a! ', b f , (/,... //, k! ', l ; ^x, ?/)*', 



the covariants in question will all be educts of (i< e. functions 

 having for their leading coefficients) the successive resultants 

 of the forms 



[(a,...h,k,l), (a'....7/ 3 #, Z')]> 

 of the forms 



[(a,..,*,*), (a f ,...h',k f )l 

 of the forms 



[(a,.-..*), (a',...h')l 



and so on, the discriminants of which may be called partial 

 resultants of the given forms ; in a word, the simplified residues 

 arising in the process of commonmeasuring in respect to one 

 of their variables two given binary quantics are differential 

 derivatives, in respect to that variable, of the educts of their 

 partial resultants (of course with the understanding that the 

 last simplified residue is the complete resultant itself). 



This seems to point to the existence of some generalized 

 statement of Sturm's theorem in which the same Educts as 



