Mr. A. Cayley on the Sections of a Quadric Surface. 43 

 the four conies may be expressed in the form 

 [~bg*-ch* + 2fyh)YZ+ 6/.ZX + cA 2 .XY=0, 



ap . YZ + (-cA 2 -«/ 2 + 2/^)ZX+ ck* . XY=0, 



aP . YZ+ bg* . ZX+(-c/ 2 -^ 2 + 2/^)XY=0, 



(_^_ c ^ + 2 /^)YZ+(-cA 2 -q/ ,2 + 2/^)ZX+(~«/ 2 -^ 2 H-2/^)XY=0, 

 which is the required form. 

 Cambridge, November 28, 1863. 



VI. Analytical Theorem relating to the Sections of a Quadric 

 Surface. By A. Cayley, Esq.* 



THE four sections x = 0, y = 0, z = 0, w = of the quadric 

 surface 



ax 2 + by 2 + 6ccy x^ab — cz 2 — dw* = 



are each of them touched by each of the four sections 

 x \/2a + y V2b±z \/c~±w Vd=0; 



where it is to be noticed that the radicals s/2a, V2b are such 

 that their product is =2 V ' ab if Vab be the radical contained 

 in the equation of the surface. There is of course no loss of 

 generality in attributing a definite sign to the radical V2a ; but 

 upon this being done, the sign of the radical V2b is determined, 

 whereas the signs of s/ c and V d are severally arbitrary. We 

 may if we please write the equation of any one of the last-men- 

 tioned sections in the form 



x s/2a + y */2b + z Vc + w */d=0, 



it being understood that the radicals V2a, V2b have each a 

 determinate sign, but that the signs of Vc and V d are each of 

 them arbitrary. 



To prove the theorem, it is enough to show (1) that the sec- 

 tions x = 0, x V2a + y VTb + z Vc + w Vd^O; (2) that the 

 sections z = 0, x */2a-\-y </2b + w\/d=0, touch each other. 



1. The sections x = 0, x V2a-\-y V2b + z Vc + w Vd=0 of 



the surface ax 1 -f by 2 -f 6xy Vab — cz 2 — du? = will touch each 

 other if, combining together the equations 



a7=0, yV%b + zV~c + wVl=O i bif—cz*—dw 2 = 0, 

 these give a twofold value (pair of equal values) for the ratios 



* Communicated by the Author. 



