Theory of Canonical Forms and of Hyperdeterminants. 409 



In consequence of the greater interest, at least to the author, 



of the preceding investigations, I have delayed the insertion of 



the promised continuation of my paper on extensions of the dia- 



lytic method, which will appear in a subsequent Number. I 



take this opportunity of correcting a trifling slip of the pen 



which occurs towards the end of the paper alluded to. The 



so u 



values of - and - become zero, and not infinite, when 7i=:=0; 



z z 



and the antepenultimate paragraph should end with the words 

 '^ an incomplete resultant." The theorem also, in the last para- 

 graph but one, should be stated more distinctly as subject to, an. 

 important exception as follows. ,,- noqjj 



Whenever the resultant of a system of equations F = 0, G=0, 

 &c. contains a factor E"", this will indicate that, on making 

 R'=0, the given system of equations will admit of being satisfied 

 by m algebraically distinct systems of values of the variables, 

 except in those cases where there is a singularity in the forms of 

 F, G, &c., taken either separately, or in partial combination with 

 one another. An example will serve to make the meaning of the 

 exception apparent. Let F, G, H denote three quadratic equa- 

 tions in X and y, so that F = 0, G=0, H = may be conceived 

 as representing three conic sections. Let R be the resultant of 

 F, G, H, and suppose the relations of the coefficients in F, G, H 

 to be such that E, = E'2 ; then ll' = will imply the existence of 

 one or the other of the three following conditions : viz. either 

 that the three conies have a chord in common, which is the most 

 general inference ; or, which is less general, that two of the 

 conies touch one another; or, which is the most special case of 

 all, that one of the conies is a pair of right lines. 



So, again, if we have two equations in x^ and their resultant 

 contains F^, this may arise either from one of the functions con- 

 taining a square factor, or from their being susceptible, on insti- 

 tuting one further condition, namely of F=0, of having a qua- 

 dratic factor in common between them. lil^mliiv^._ 



Lincoln's-Inn-Fields, (p^ '^^^^ 



October 14, 1861. : ^ hri»iii;i^ ^'lour an'fi if 



P.S. Tlie conjecture made in the preceding pages has been 

 since confirmed by the discovery of a modification in the ca- 

 nonical form applicable to functions of the sixth degree, which 

 simplifies the theory in a remarkable manner. Assume /(^,y) 

 a function of the 6th degree as equal to 



ai^ + hv^ 4- cw^ + muvw . (z« — ?;) (v — w) {w — w)^^^,^ ^ __v *^ ^ 

 where u, v, w, linear functions of x and y, satisfy the equatiott ./ 



