68 
ON A SURFACE OF THE FOURTH ORDER. 
[313 
/ 
The points M, M' of the curve, which are images each of the other in respect to 
the circle through A, B, G, may be called conjugate points of the curve. The above- 
mentioned circle, the intersection of the three spheres, is the circle having MM' for 
its diameter; hence the required surface is the locus of a circle at right angles to the 
principal plane, and having for its diameter MM', where M and M' are conjugate 
points of the curve. 
In the particular case where the equation of the surface is 
BG.AP+GA.BP + AB.GP = 0, 
the principal section is the circle through A, B, C, twice repeated. Any point on the 
circle is its own conjugate, and the radius of the generating circle of the surface is 
zero; that is, the surface is the annulus, the envelope of a sphere radius 0, having 
its centre on the circle through A, B, G. Or attending to real points only, the surface 
reduces itself to the circle through A, B, G. But this last statement of the solution is 
an incomplete one. The equation of an annulus, the envelope of a sphere radius c, 
having its centre on a circle radius unity, is 
V# 2 + y 1 = 1 ± Vc 2 — z 2 ; 
and hence putting c = 0, the equation of the surface is, 
V# 2 + if = 1 + zi 
(if, as usual, ¿ = V — 1), or, what is the same thing, it is 
a- 2 + y 2 + (z ± i)- = 0 ; 
that is, the surface is made up of the two spheres, passing through the points A, B, G, 
and having each of them the radius zero; or say the two cone-splieres through the 
points A, B, G. In other words, the equation 
BG.AP + GA .BP + AB.CP = 0 
is the condition in order that the four points A, B, G, P may lie on a sphere radius 
zero, or cone-sphere. Using 1, 2, 3, 4 in the place of A, B, C, P to denote the four 
points, the last-mentioned equation becomes 
12.34+13.42 + 14.23 = 0; 
and considering 12, &c. as quadratic radicals, the rational form of this equation is 
□ = 
0 , 
12 ! 
2 
13, 
2 
14 
21 2 > 
0 , 
23*, 
24’ 
3i 2 , 
82* 
0 , 
2 
34 
41 2 , 
42 2 , 
43 2 , 
0