186 Prof. J. J. Sylvester's Proof of the undemonstrated 



be called the outstanding factor. Of this I shall proceed to 

 speak. 



Let x be replaced by hx-\-ly, 



y „ „ kx + my. 



Then the outstanding factor for the transformed D (a pure 

 differential in x of the grade 8) may be proved by repeated 

 applications of Salmon's theorem to be equal to 



( 1+ Ao. + (A) 2 ft + ...W 



\ m \mJ 1.2 / 



where of course the series of terms in the development will, 

 after the (8 + l)th term, vanish spontaneouslv- In other words, 



KO. 



the outstanding factor of the transformed D is m s e '» .D, 

 where it will be noticed that only the coefficients of substitu- 

 tion due to the change in y make their appearance. 



If now we take any mixed differential, say the qth provect 

 of D, i. e. O q . D, its outstanding factor, I find, will be the qth 

 emanant of the outstanding factor for D, i. e. will be 



(**>£>'.{•**)* 



And here for the present I end. The subject is, as it was, a 

 vast one; and this conception of mixed differentiants opens out 

 still vaster horizons. Every thing grown on American soil or 

 born under the influence of its skies, as its lakes, its rivers, its 

 trees, and its political system, seems to have a tendency to rise 

 to colossal proportions. 



I will merely add one remark which has occurred to me 

 relating to Sturm's theorem and the process of Algebraical 

 common measure in general. If f(x, y) be a rational integral 

 function of x, y, '<mdf(x, y) its derivative in respect to x, and 

 we perform the process of common measure between them 

 regarded as functions of x, we know that the irreducible part 

 of the successive remainders taken in ascending order, say 

 U , Ui, U 2 , . . . , will have for their leading coefficients (say 

 D^D^Dg...), the discriminants of / and of its successive 

 derivatives in respect to x respectively. 



Here D is an invariant of the given form ; 



D 1 (a differentiant in x) will be the leading coefficient 

 of the covariant 



