[183 
184] 
115 
' 2 ) = 0. 
__tre of 
ntre of 
of the 
receded 
nilitude 
is the 
184. 
A THEOREM RELATING TO SURFACES OF THE SECOND ORDER. 
[From the Quarterly Mathematical Journal, vol. II. (1858), pp. 140—142.] 
Given a surface of the second order 
(a, h, c, d, f, g, h, l, m, n) (x, y, z, w)- = 0, 
and a fixed plane 
imagine a variable plane 
ax + ßy + <yz + 8w = 0, 
%x + iiy + £z + cow = 0, 
subjected to the condition that it always touches a surface of the second order, or 
what is the same thing such that the parameters 77, £ co satisfy a condition 
(a, b, c, d, f, g, h, 1, m, n) (f, 77, £ &>) 2 = 0. 
The given surface of the second order, and the variable plane meet in a conic, 
and the fixed plane and the variable plane meet in a line, it is required to find the 
locus of the pole of the line with respect to the conic. 
The pole in question is the point in which the variable plane is intersected by 
the polar of the line with respect to the surface of the second order: this polar is 
the line joining the pole of the fixed plane with respect to the surface of the second 
order, and the pole of the variable plane with respect to the surface of the second 
order. Let a 1} /3 1} y 1} S 1} be given linear functions of a, /3, 7, 8, and f, 77, £ co, be 
given linear functions of f, 77, £, co, viz., if 
(21, 23, G, 2), g, @, $, 8, m, n 
15—2