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VII. A Memoir on Cubic Surfaces. By Professor Cayley, F.B.S. 
Received November 12, 1868, — Read January 14, 1869. 
The present Memoir is based upon, and is in a measure supplementary to that by Pro- 
fessor 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. cliii. (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, attend- 
ing 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-coor- 
dinates of the several surfaces (orders 12 to 3) reciprocal to these repectively. 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 ” 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 
through the Memoir. 
The twenty-three Cases of Cubic Surfaces — Explanations and Table of Singularities. 
Article Nos. 1 to 13. 
1. I designate as follows the twenty-three cases of cubic surfaces, adding to each of 
them its equation : 
I —12, 
II =12 — C 2 , 
III =12— B 3 , 
IV =12— 2C„ 
V =12— B 4 , 
VI =12— B 3 — C 2 , 
VII=12— B 5 , 
(X, Y, Z, W) 3 =0, 
W (a, b, c,f g, hJX, Y, Z) 2 +2£XYZ=0, 
2W(X+Y+Z)(ZX+mY+nZ)+2£XYZ=0, 
WXZ+Y 2 (yZ+SW)+(«, b , c, dJX, Y) 3 =0, 
WXZ+(X+Z)(Y 2 -aX 2 -6Z 2 )=0, 
WXZ+Y 2 Z+(«, b, c, dJX, Y) 3 =0, 
WXZ+Y 2 Z+YX 2 — Z 3 =0, 
