360 Mr Bromwich, The Classification of Conies and Quadrics. 

 Similarly 



DC=-E + ^ 





Pr 

 



Expand D in powers of (1/X), and let us write 



D = 2D m X™ 



where m will extend from — 1 to — x> and may also include a 

 finite number of positive indices as well. 



Write further, also expanding in powers of (1/X), 



l_ 



a; 



Ps, 



— ^r] m K , 





Pr 







= 2f m \ m , 



then by the symmetry of C, % m will be the same function of 

 x x , ... , x n that r) m is of y l , ..., y n . We shall suppose that A.* -1 

 is the highest power of X which occurs in %f; m \ m ; and X* 3 the 

 highest in 2D m X m . Then we have 



(\A-B)2D m V n (m=ft£-l, ...,-») 



= -E+p x {^ Vm \ m ) (m = a-l, a-2,...,-oo). 



Hence /3?(a- 2), and if /3 > a — 2 we shall have ^4^ = 0, 

 so that ' A | = if Dp 4=0. By comparing coefficients of other 

 powers of X, we find in general 



■A--*-'m—i -BUm ~ PxVmi 



but for m = 0, A D_ x - BD = -E + p xVo . 



It is to be noted that the products on the left are all 

 symbolical ; while those on the right give a simple product of 

 two linear functions which will be itself a bilinear form, capable 

 of symbolical combinations. 



Similarly, from the value of DC, we find 



D m -i-A - D m B = p y % m , in general, 



J) X A -D B = -E + p y % (m = 0). 



Hence 



AD_,A = A (D^A - D B) + (AD - BD X ) B + BD,B 



= -A + A (p y % ) + (p xVl ) B + BD,B, 



where now A (p y %o) and (p x Vi) B are symbolical products ; to form 

 say the former of them, we have to replace x r in (p y %o) by the 



