362 Mr Bromwich, The Classification of Conies and Quadrics. 

 where we write 



F=tF m \™= 7 -\ c rs , p r , Xb rt x t 



p s , 0, 



%b ts x t , 0, 



, (ra = /3,/3-l, ...,-x), 



G = XG m \ m = 



A, 



t 



p s> o, 



Za ts x t , 0, 



r=S£ m \™ = — C rs , Pr 



Xb ts x t , 



,(m = /3,/3-l,...,-x), 



, (m = a — 1 , a — 2, . . ., — x ), 



8 = 2S m X w = A /A 1} (m = a, a - 1, ...,.- oo ). 



2. Application to the principal axes problem. 



Let 4 = be the equation to a quadric in homogeneous 

 coordinates of space of (n— 1) dimensions; and then _B = will 

 be chosen to be any hyper-sphere in this space, the coefficients of 

 B being adjusted so that for a point x not on the hyper-sphere, B 

 will represent the square of the tangent from x to the hyper- 

 sphere. The coordinates of any point will be connected by a 

 relation, which is here assumed to be 



r 



so that the hyper-plane at infinity is p x = 0. 



Now returning to the forms found at the end of Section 1, we 

 see that we can reduce O to the form 



G = -ty^/e r (X-e r ), (r = l,2,...,h), 



by using the process given by Weierstrass 1 , Darboux 2 , and 

 Stickelberger 3 . Here (\ — r ) is a typical factor of A 1} y r is a 

 corresponding linear function of the x's, and h is the index of the 

 highest power of X in A x . There may also be positive powers of A, 

 in the expression for G, but we are not concerned with these. 



1 Berl. Berichte, 1868 = Ges. Werhe, 2, p. 19. 



2 Liouville (1874), 19 (2nd Ser.), p. 347. 



3 Grelle (1879), 86, p. 20. 



