122 Mr. J. J. Sylvester on Polynomial Functions which 



dratic function ; the speciality in the latter case consisting merely 

 in the fact that v being equal to n, the rectangular matrix 

 becomes a square, and there is only one full determinant. 

 Moreover, for any other value of (r) the above rule coincides with 

 that given by me some time back in the Philosophical Magazine 

 for the case of quadratic functions. 



Professor Hesse's rule for finding conditions applicable to the 

 loss of one order is, as I have already stated, a consequence of 

 the more simple scheme of conditions above given. It consists 

 in forming the determinant 



dx^^ dx^.dx^* ' * * dx^.dcCn 

 d^V d^V d^ 



dx^.dic^' dx^.da;^ * • • • d^^^dx^ 

 d'^V d'^V d^V 



dx^,dx^* dx^.dx^' ' * ' dx„.dx„* 



and equating the coefficients of this determinant fully developed 

 separately to zero*. The attachment of the Professor to this 

 particular form of covariant (I use the language of the calculus 

 of forms) is readily intelligible, seeing the admirable application 

 which he has made of it to the canonization of the cubic func- 

 tion of 3 variables, but it is really foreign to the nature of the 

 present question ; the coefficients of this covariant may easily be 

 shown to be merely the full determinants of the nxv rectangular 

 matrix above described, or linear functions of these said deter- 

 minants with numerical coefficients. Hence the ground of its 

 applicability. 



Returning to the rule of the matrix, if we suppose the number 

 of variables to be 2, and call the coefficients of U 



n-l 

 «o, wflp n . —^ a^ . . ,an, 



our rectangle becomes 



* A form capable of being so derived I have elsewhere termed (in com- 

 pliment to M. Hesse) the Hessian of the function to which it appertains. 

 This is the trivial name which is much needed on account of the frequent 

 occurrence of the form, and has been adopted by Mr. Salmon in his admi- 

 rable treatise on the higher plane curves. In systematic nomenclature it 

 would be termed the discriminant of the quadratic emanant, or more briefly, 

 the quadremanative discriminant. I have discovered quite recently that 

 the long sought for symmetrical, and by far the most easy practical process 

 for discovering the number of the real roots of an equation, is contained 

 in, and may be deduced immediately from, a certain transformation of its 

 Hessian! 



