&c. [331 
332] 
133 
rough the three 
per for putting 
ANALYTICAL THEOREM RELATING TO THE SECTIONS OF A 
QUADRIC SURFACE. 
0, 7=0, Z = 0, 
Drill 
[From the Philosophical Magazine, vol. xxvn. (1864), pp. 43, 44.] 
.17=0, 
The four sections x — 0, y = 0, z = 0, w = 0 of the quadric surface 
.17=0, 
)XY = 0, 
ax 3 + by- + 6xy Vab — cz- — dw 2 = 0 
> 17 = 0, 
are each of them touched by each of the four sections 
# V2a + y V26 + .z Vc + w Vd = 0; 
where it is to be noticed that the radicals V2a, V2b are such that their product is 
= 2Va6 if Va6 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 Vc" and VcZ are severally arbitrary. We may if we please write the equation of 
any one of the last-mentioned sections in the form 
x V2a + y V26 + 2 Vc + tu Vd = 0, 
it being understood that the radicals V2a, V2b have each a determinate sign, but that 
the signs of Vc and Vd are each of them arbitrary. 
To prove the theorem in question, it is enough to show (1) that the sections x — 0, 
x V2a+y V26 + ^Vc + w Vd= 0; (2) that the sections z = 0, x V2a + y V26 + w Vd= 0, 
touch each other.