[901 
902] 
51 
2)log«; 
£ log x. 
■) ~ i log «• 
formula 
902. 
ON THE FOCALS OF A QUADRIC SURFACE. 
2 K 
"'7- 
indefinitely near to 1; 
near to 1, say the value 
d K' = ^7t, and therefore 
ition between a and ¡3; 
-4**?. = - i log «• 
7T 
log <*, 
coefficient — \\ but there 
¡onstant '577.... 
however small a may be 
ge, not only mra is not 
the general term of the 
l somewhat more rapidly 
bive value less than but 
rm approximates to —, 
[From the Messenger of Mathematics, vol. xix. (1890), pp. 113 117] 
In plane geometry, the focus of a curve is the node of the circumscribed line- 
system of the curve and the circular points at infinity; and so, in solid geometry, 
the focal of a surface or curve is the nodal line of the circumscribed developable 
of the surface or curve and the circle at infinity. And as in plane geometry the 
circular points at infinity may be regarded as an indefinitely thin conic, so in solid 
geometry the circle at infinity may be regarded as an indefinitely thin quadric surface. 
In plane geometry, let it be proposed to find the circumscribed line-system 
(common tangents) of the two quadrics 
if a common tangent be 
then we have 
here writing for shortness 
we have 
i+f + i-o. 
a b c a 1 
•%■£+?-<>; 
b c 
%x+ yy +£z = 0, 
at? + bf + c? = 0, 
a'if + b'rf + c'£ 2 = 0; 
/, g, h = bd-b'c, ca'-c'a, ab'-a'b, 
s : V : =/ : 9 
and thence the tangent is 
®V(/) ± y*J(9) ± *№) = 0, 
viz. we have thus four tangents, and the rationalised form is of course 
/ 2 « 4 + g-y 4 + h 2 z i — 2 ghy 2 z 2 — 2 hfz 2 x 2 — 2fgx 2 y 2 = 0. 
7- 
■2