184] A THEOREM RELATING TO SURFACES OF THE SECOND ORDER. 117 
the last preceding system of equations may be written 
£ = Ax + fjiOL, 
v =Xy + /¿0, 
£ = \z + fx y, 
co = Aw -f- /x8, 
equations in which A, ¡x are indeterminate, and where x, y, z, w may be considered 
as current coordinates, and this system represents the polar above referred to. Com 
bining the equations in question with the equation 
£x + yy + & + coiv = 0, 
of the variable plane, we have 
A (xx + yy + zz + ww) +/x (ax + ¡3y + yz + 8iu) = 0, 
i.e. 
A (a,...) (x, y, z, w)' 2 + y (ax + ¡3y + yz + 8w) = 0, 
or what is the same thing 
A : fx = ax + ¡3y + ryz + 8w : (a,...)(x, y, z, w) 2 , 
and substituting these values in the expressions for £, 77, £, co we have £, 77, £, co in 
terms of the coordinates x, y, z, w of the pole above referred to, i.e., if for shortness, 
U = (a, b, c, d, f g, h, l, m, n) (x, y, z, w) 2 , 
P = ax + fiy + yz + 8w, 
then 
£ = \Pd x TJ- aU, 
77 =%Pd y U-0TJ, 
£ = \Pd z U — yU, 
co = lPd w U- 8U, 
and combining with these equations the equation 
(®v0 (£ v, £> «) 2 = 0, 
(a...) (\PdJJ-aU, \Pd y U-QP, \PdJJ-yU, \PdJJ-8U) 2 = 0, 
for the required locus of the pole of the line of intersection of the variable plane 
and the fixed plane, with the conic of intersection of the given surface of the second 
order and the variable plane. The locus in question is a surface of the fourth order; 
and it may be remarked that this surface touches the given surface of the second 
order along the conic of intersection with the fixed plane. 
7 tli April, 1857.