Mr. A. Cayley on a Surface of ihe Fourth Order. 493 
so the circle through M, in respect to which C, A are images 
each of the other, and the circle through M, in respect to which 
A, B are images each 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, C, 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, C are points of this circle, or we have 
MA MB MC” 
WA MB MC? 
that is, the equation : 
AM +»BM+rvCM=0 
being satisfied, the equation 
VAM! + wBM!'+vCM'=0 
is also satisfied. 
The points M, M! of the curve, which are images each of the 
other in respect to the circle through A, B, C, 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 
BC.AP+CA.BP+AB.CP=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,C. Or attending to real 
points only, the surface reduces itself to the circle through 
A, B, C. But this last statement of the solution is an incom- 
plete one. ‘The equation of an annulus, the envelope of a sphere 
radius c, haying its centre ona circle radius unity, is 
Vxrty=l+ V¥e—2?; 
and hence putting c=0, the equation of the surface is, 
Va? +y2=ltzi 
