412] 
359 
412. 
A MEMOIR ON CUBIC SURFACES. 
[From the Philosophical Transactions of the Royal Society of London, vol. clix. (for the 
year 1869), pp. 231—326. Received November 12, 1868,—Read January 14, 1869.] 
The present Memoir is based upon, and is in a measure supplementary to that 
by Professor Schlafli, “ On the Distribution of Surfaces of the Third Order into Species, 
in reference to the presence or absence of Singular Points, and the reality of their 
Lines,” Phil. Trans, vol. CLlii. (1863), pp. 193—241. But the object of the Memoir is 
different. I disregard altogether the ultimate division depending on the reality of the 
lines, attending only to the division into (twenty-two, or as I prefer to reckon it) 
twenty-three cases depending on the nature of the singularities. And I attend to the 
question very much on account of the light to be obtained in reference to the theory 
of Reciprocal Surfaces. The memoir referred to furnishes in fact a store of materials 
for this purpose, inasmuch as it gives (partially or completely developed) the equations 
in plane-coordinates of the several cases of cubic surfaces, or, what is the same thing, 
the equations in point-coordinates of the several surfaces (orders 12 to 3) reciprocal 
to these respectively. I found by examination of the several cases, that an extension 
was required of Dr Salmon’s theory of Reciprocal Surfaces in order to make it 
applicable to the present subject; and the preceding “Memoir on the Theory of 
Reciprocal Surfaces,” [411], was written in connexion with these investigations on Cubic 
Surfaces. The latter part of the Memoir is divided into sections headed thus:— 
“Section 1 = 12, equation (X, Y, Z, W) 3 = 0 ” &c. referring to the several cases of the 
cubic surface ; but the paragraphs are numbered continuously throughout the Memoir. 
Article Nos. 1 to 13. The twenty-three Cases of Cubic Surfaces—Explanations and Table 
of Singularities. 
1. I designate as follows the twenty-three cases of cubic surfaces, adding to each 
of them its equation: 
I =12, (X,Y,Z,W) 3 = 0, 
II =12-02, W(a, b, c, f g, h\X, Y, Z)' 2 + 2kXYZ = 0, 
III =12 — R i} 2 W (X + Y + Z) {IX ni 1 + nZ) + 2kXYZ = 0