368 Mr Bromwich, The Classification of Conies and Quadrics. 



every minor up to the Ith minors, and we shall have corresponding 

 parts of J., B in Darboux's form 



A = c(V 1 2 +...+ Vf), 



B= (VS+... + V?), 



where V k is the limit for A = c of the function 



and 



here the q's are arbitrary constants and the value zero for c is not 

 excluded. 



But with regard to the infinite root of \A — XB\ = we shall 

 have to take special steps. I apply a method given by myself 

 in the note quoted above (p. 358). Let us suppose first of all 

 that A n_1 is the highest power of A that occurs in \A — \B\, and 

 then write U for the term independent of A in the expansion in 

 powers of (1/A) of the expression 



(- A A)* 



A s 



, (r, 8 = 1, 2, ...,n), 







where we put A s = a u v x + . . . + a m v n . 



With this notation we have the term U 2 in A and no corre- 

 sponding term in B, by the results of the note quoted. Now the 

 terms of highest degree in A are respectively (since the minor of 

 b rs in | B | is kp r p s ), 



in A , (- A)' 1-1 Xa rs kp r p s = k{- A)' 1 " 1 %a rs p r p s , 



in A„ - (- A)' 1 " 1 %< V J) (%•#>) = ~k(- X)*" 1 (Xp r qr {1) y, 



in 



a rs — \b rs , q r {l 

 A s , 



-k(-\T-i(2p r q r U)(tp r A r ). 



Thus we find U 2 = (%>,.J. r ) 2 /(2a,. s p,.j9 s ). 



But Ave are to have that for the special values v r = p r , V is to 

 be unity (for %p r sc r = 1). For these special values V lt ..., V n _ 1 are 



r 



zero because B s — 0; and then %p r A r = H,a rs p r p s . 



