370 Mr Bromwich, The Classification of Conies and Quadrics. 



If then we write 



«iF (- &/&)i = U 2 -^U 1 (« 2 / a] + &/&), 

 we shall have V = 1 when w r = p,.. 



Now we find 



1 , . , , '■_ \ 



I ^ s o| 



= XF 1 2 + 2F F 1 (-« 1 /^ + .... 



According to the results in the note already quoted we now 

 see that there is a term V^ in B and 2I 7 "P 1 (- ajkf in A Hence 

 the reduced forms are 



A = 2V V 1 (-a 1 /kf + Xc r V t ? 



(r=2,S,...,n-l). 



r 



Here again, by the form of B and the fact that V = 1 corre- 

 sponds to v r = p r , we see that the point-coordinates given by 



V + V 1 X 1 + . . . + Vn^Xn^ = V X X^ + . . . + V n X n 



are rectangular Cartesians. Hence we have the quadric referred 

 to its principal planes. 



We have then the classification : 



Conies (n = 3). 

 (i) 5a rs p r p s 4= 0. 



(1) A = avf + bv 2 2 + (ZttrsPrPs) v 2 , central conic 



(ellipse, hyperbola or circle). 



(2) A = av^ + (Zarsprps) v 2 , pair of points- 

 (ii) ta rs p r p s = 0. 



(3) A = av£ + 2v Q v 1 (— a/k)*, parabola. 



a = Lt (- A>/\) 



It will be observed that the cases (2), (4), (5), (6), (7) of the 

 previous classification do not appear here, the tangential equation 

 being degenerate ; on the other hand, case (2) above does not 

 appear in the previous list, as then the point-equation is de- 

 generate. 



