Mr Bromwich, The Classification of Conies and Quadrics. 369 



Thus we write 



V = (%p r A r )/(Xa rs p r p s ), 



and then we have the term V 2 (Za rs p r p s ) in the reduced form of 

 A, with no corresponding term in B. Hence we shall have the 

 complete forms 



A = 1c r V r 2 + (2a rs p r p s ) F 2 , (r = 1, 2, . . ., n - 1), 



r 



B = 2V r \ 



r 



From these it is obvious (remembering the special meaning 

 of B) that the point-coordinates given by 



V +V 1 X 1 +...+ V n _! Xn_j = ViOOj. + ...+ v n x n 



are rectangular Cartesians. We have accordingly found the 

 principal planes of the quadric, and we can put down by inspection 

 the lengths of the principal axes. 



In the next place we may have X, = oo a repeated root of 

 \A—\B\, then the highest power of X in | A — XB | is A," -2 . The 

 condition for this is at once seen to be Xa rs p r p s = 0. Let us 

 then write 



A = (-\)- 2 [a ] +a 2 /\ +...], 



A, = (-X.) w - 1 [/3 1 +&/X +...], 



^ = -k(lp r q r wy ; ) 



a rs - \b rs , q r m I = (- \y-i [U 1 +U 2 /\ + ... ]. 

 A s , 



(so that 



Then expanding — 



AA 



a rs — \b rs , q r m | 2 in powers of (1/X), 

 A s , 



we find 



A, 

 <hfr 



■ / ' + ^--wS + l)H + 



and so write U 1 = V x (a.^)*. We must next calculate the values 

 of U l , U 2 when v r =p r . We have that if v r =p r 



and so 



(- X)' 1 - 1 [ U x + TJJk + ...]=- A O^frW), 



0i =0, tf 2 = a, (X M /» ) = «:(- A/*)*. 



