admit of reduction in the number of Variables, 123 



flg 



«n-: 



and the conditions become 

 Oq. a^— ai^=0 



flg — ffi 



fl2=±0 



.«. 



.-,=0 



.««-i = 0, &c., 



all of which equations are obviously true (when the function 

 loses an order, that is to say, becomes a perfect power) and are 

 satisfied (special cases excepted) when any {n—1) independent 

 equations out of the entire number obtain ; so that the number 

 of conditions implied in the property to be represented is in 

 exact conformity with the number of independent equations 

 derived from the matrix, i. e. equations which, when satisfied, 

 will in general cause all the rest to be satisfied. This confor- 

 mity manifests itself also in the case of a quadratic function of 

 n variables. But except in these two limiting (and, in an occult 

 sense, reciprocal*) cases of a function of 2 variables of the /ith 

 degree, or of the degree 2 and n variables, this conformity in 

 measure as the degree or number of variables rises, although it 



* There are frequent cases occurring in the calcuhis of forms of inter- 

 change between the degree of a function and the number of variables which 

 it contains. Thus, to select a striking example (although one where the 

 interchange is not exact), the theory of the real and imaginary roots or 

 factors of a homogeneous function of 2 variables and of the nth degree 

 may be shown to be immediately dependent upon the determination of the 

 specific nature of a concomitant homogeneous function of the 2nd degree 

 and of (w — 1) variables. For instance, if any ordinary algebraical equation 

 of the 5th degree be given, a homogeneous quadratic function of 4 variables 

 may be constructed, representing, consequently, a surface of the 2nd degree 

 [the coefficients of which (as indeed is true whatever be the degree of the 

 equation) will be quadratic functions of the coefficients of the given equa- 

 tion] ; and such that, according as the surface so represented belongs to the 

 class of 1°, impossible surfaces; 2°, the ellipsoid or hyperboloid of two 

 sheets ; 3°, the hyperboloid of one sheet ; the given equation will have 5, 

 3, or only 1 real root ! Moreover, an equality between two of the roots of 

 the equation will be denoted by the loss of one order in the associated 

 (quadratic function ; and so many orders altogether will be lost as there are 

 independent equalities existing between the roots. An entirely new light 

 is thus thrown on M. Sturm's theorem ; and the number of real and imagi- 

 nary roots in an equation is for the first time made to depend upon the 

 signs of functions symmetrically constructed in respect to the two ends of 

 the equation, which has long been felt as a desideratum. 



K2 



