-^ . ^ .- [ 391 ] ' ■ '* 'm 



LX. On a remarkable Discovery in the Theory of Canonical Forms 

 and of Hyper determinants. By J. J. Sylvester, M.^.,F.i2.>S.* 



IN a recently printed continuation of a paper which appeared 

 in the Cambridge and Dubhn Mathematical Journal f^ I 

 published a complete solution of the following problem. A 

 homogeneous function of Xj y of the degree 2n-\-l being given, 

 required to represent it as the sum of n-\-l powers of linear 

 functions of x, y. I shall prepare the way for the more remark- 

 able investigations which form the proper object of this paper, 

 by giving a new and more simple solution of this linear trans- 

 formation. 



Let the given function be . ... ^l^^ ,, .. 



fli.«2«+i ^ ^2n + l)ai,se^y+{2ni-l)^^^~aci.x^''-'^ y, &c. 



+«2«+2.y^'*"^S 



and suppose that this is identical with 



The problem is evidently possible and definite, there being 

 271 + 2 equations to be satisfied, and (2w + 2) quantities pi, q^, 

 &c. for satisfying the same. 



In order to effect the solution, let 



qi=Pi'\ .^. .^^.. 



q2-P2'\ 

 &c.= &c. 



Eliminate p^, pc^...pn+i between the 1st, 2nd, 3rd...(n + l)th 

 equations, and it is easily seen that we obtain 



««+ 1 •— «„SXi + an-iZ\'K2 &c. + ajXjXg . . . \n+ 1 = 0. 



* Communicated by the Author. 



t Published under the title of *An Essay on Canonical Forms/ by Bell, 

 Fleet Street. . 



