ON A SURFACE OF THE FOURTH ORDER. 
67 
[313 
313] ON A SURFACE OF THE FOURTH ORDER. 67 
each of the other; that is, the centre of the sphere lies on the line BC, and the 
product of its distances from B and G is equal to the square of the radius; in like 
manner the second sphere is such that, with regard to it, G and A are the images 
each of the other; and the third sphere is such that, with regard to it, A and B 
are the images each of the other. The three spheres intersect in a circle through M 
at right angles to the principal plane (that is, the three spheres have a common circular 
section), and the equations of this circle may be taken to be 
AP BP CP 
AM ~ BM ~ CM' 
DER. 
It is clear that the circle of intersection lies wholly on the surface. 
The spheres meet the principal plane in three circles, which are the diametral 
circles of the spheres ; these circles are related to each other and to the points A, B, G, 
in like manner as the spheres are to each other and to the same points. The circles 
have thus a common chord; that is,' they meet in the point M and in another 
L—495.] 
point M': and MM' is the diameter of the circle, the intersection of the three spheres. 
bure of the surface, 
It may be shown that M, M' are the images each of the other in respect to 
the circle through A, B, C. In fact, consider in the first place the two points 
A, B, and a circle such that, with respect to it, A, B are the images each of the 
on the ratios only 
other; take M a point on this circle, and let 0 be any point on the line at right 
angles to AB through its middle point, and join OM cutting the circle in M'; then 
it is easy to see that M, M' are the images each of the other, in regard to the 
circle, centre 0 and radius OA (= OB). Hence starting with the points A, B, G and 
iously symmetrical 
say the principal 
tat 
the point M, let 0 be the centre of the circle through A, B, C, and take M the 
image of M in respect to this circle; then considering the circle which passes through 
M, and in respect to which B, G are images each of the other, this circle passes 
through M'; and so the circle through M, in respect to which G, A are images each 
of the other, and the circle through M, in respect to which A, B are images each 
his Higher Plane 
emarked that the 
ich is therefore of 
at each of these 
>air intersecting a 
it there are four 
rcle with A, B, G, 
ts A, B, C, D, the 
e form as it is in 
of the other, pass each of them through M; that is, the three circles intersect in M'. 
It is to be noticed that M', being on the surface, must be on the principal 
section; that is, the principal section is such that, taking upon it any point M, and 
taking M' the image of M in regard to the circle through A, B, G, then M is also 
on the principal section. It is very easily shown that the curve of the fourth order 
possesses this property; for M, M' being images each of the other in respect to the 
circle through A, B, C, then A, B, G are points of this circle, or we have 
MA MB MG 
M'A ~ M'B ~ MG ’ 
tions 
that is, the equation 
XAM + gBM + vGM = 0 
being satisfied, the equation 
\AM' + gBM' + vCM' = 0 
ough the point M. 
0 are the images 
is also satisfied. 
9—2