Mr Bromwich, The Classification of Conies and Quadrics. 361 



linear function Xa tr x t and we thus obtain j>y^o where £„' is the 

 term independent of \ in the expansion of 



Ax 



Za ts % t , 



In like manner (p x r)j)B =p x ij 1 ", where 

 1 



Ai 



= v>. 



We shall use % m ", 97™ in analogy with r} m ", % m ' respectively, and 

 we now find 



while 



A = BD,B - AD^A + Py £ Q ' +p xVl ", 

 B = BD B - AD^A +p y %- 1 '+p x Vo". 



It must be observed that £ m ', lj m " are not independent : for 

 we have 



\(2£ m '\™)-ZZ m "\ m = 



A: 



Crs) Pr 

 XCtgOBt, 



Pa 



V 



where we write 



A = I c r 



From these equations it appears that when Aq/Aj is expanded 

 in powers of (1/X.) the highest power of A, will be \ a ; further, 

 if we write 



A /A 1 = S a V + S a _ 1 A,*- 1 +..., 

 we shall have 



So— 1 = Px®a> 



and generally &n-i - %m = - £>Ai- 



Substituting, we have 



A = BD^B - AD_,A +p y Z" + p x v" -p x p y $i, 

 B = BD B - AD_ 2 A + p y % " + p xVo " '- p x pyS . 



These forms will be the best for our future investigation; and 

 now write y r — x r in each term, which will give the corresponding 

 quadratic forms 



A =F 1 -G- 1 + 2p x £ 1 -p x *8 1 , 



B = F -G- 2 + 2p x Z Q - Pa ?8 , 



