188 On the Fundamental Theorem of Invariants. 



above referred to shall appear, but where, instead of their deri- 

 vatives in respect to one of the variables being made use of, 

 perfectly general Emanants of them shall be employed as the 

 Criterion functions. For I need hardly add that all Educts 

 (although not written so as to show it in what precedes) are in 

 fact symmetrical in respect to the two sides of the quantic to 

 which they belong. 



On various a priori grounds I suspect the generalized 

 theorem to be as follows. If X 2ja is the covariant (of degree 

 2/ub) whose fjuth derivative in respect to x is a Sturmian Auxiliary 

 Proper to F(x, y), we may substitute throughout for all the 

 values of //-, instead of each such derivative, the more general 



one (f-j g — J X 2/x , where /and gare any assumed positive 



constants, of course with the understanding that the second 



criterion also is to be If '-= a —-)f'va. lieu of -r — And the 



V dx u dy/ J dx 



method of Sturm will still be applicable for finding the posi- 



x 

 tions of the real roots of - in f(x, y) = when we use these 



<y 



more general derivatives as the criteria instead of Sturm's 



own. When g = the theorem is that of Sturm ; when/=0 



it is an immediate deduction from this theorem applied to 



finding the positions of the root values of-, when it is borne 



x v 



in mind that the motions of - and of -, as regards ascent and 



y x 7 & 



descent (excluding the moment for which either of these.ratios 

 is indefinitely near to zero) are inverse to each other. It is 

 this that accounts for the negative sign which precedes g. 



It is difficult to conceive by what theorem other than the 

 assumed one the chasm between those extreme cases can be 

 bridged over; and all analogy and all belief in continuity vetoes 

 the supposition that no such bridge exists. " Divide et impera" 

 is as true in algebra as in statecraft ; but no less true and 

 even more fertile is the maxim " auge et impera. n The more 

 to do or to prove, the easier the doing or the proof. 



November 19, 1877. 



