66 
[313 
313. 
ON A SURFACE OF THE FOURTH ORDER. 
[From the Philosophical Magazine, vol. xxi. (1861), pp. 491—495.] 
Let A, B, C be fixed points; it is required to investigate the nature of the surface, 
the locus of a point P such that 
\AP + pBP + vGP = 0, 
where X, p, v are given coefficients ; the equation depends, it is clear, on the ratios only 
of these quantities. 
The surface is easily seen to be of the fourth order; it is obviously symmetrical 
in regard to the plane ABC; and the section by this plane, or say the principal 
section, is a curve of the fourth order, the locus of a point M such that 
\AM + pBM + vCM = 0. 
The curve is considered incidentally by Mr Salmon, p. 125 of his Higher Plane 
Curves [Ed. 3, p. 126 and see also p. 240 et seqi]; and he has remarked that the 
two circular points at infinity are double points on the curve, which is therefore of 
the eighth class. Moreover, that there are two double foci, since at each of these 
circular points there are two tangents, each tangent of the one pair intersecting a 
tangent of the other pair in a double focus; hence, further, that there are four 
other foci, the points A, B, C, and a fourth point D lying in a circle with A, B, C, 
and which are such that, selecting any three at pleasure of the points A, B, G, D, the 
equation of the curve is in respect to such three points of the same form as it is in 
regard to the points A, B, C. 
Consider a given point M, on the principal section, then the equations 
BP _ CP CP_ _ AJP AP _ BP 
BM ~ CM’ CM~ AM y AM~ BM 
belong respectively to three spheres: each of the spheres passes through the point M. 
The first of the spheres is such that, with respect to it, B and C are the images