1869.] 



Prof. Cay ley on Cubic Surfaces. 



221 



B, C, are of course also considered. An equation of Dr. Salmon's is 

 presented in the extended form, 



a'=4n(n— 2)-86-llc— 2/-3 x '-2C'-4B' ; 



and it is remarked that <r' denotes the order of the spinode-curve. The 

 Memoir contains an entirely new formula giving the value of j3', but some of 

 the constants of the formula remain undetermined. 



III. " A Memoir on Cubic Surfaces." 

 By Professor Cayley, F.R.S. Received November 12, 1868. 



(Abstract.) 



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. cliii. (1863) pp. 193-241. 

 But the object of the Memoir is different. I disregard altogether the ulti- 

 mate 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 de- 

 veloped) 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 Reci- 

 procal 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 principal results are included in the following Table of singularities. 

 The heading of each column shows the number and character of the case 

 referred to, viz. C denotes a conic node, B a biplanar node, and U a 

 uniplanar node ; these being further distinguished by subscript numbers, 

 showing the reduction thereby caused in the class of the surface : thus 

 XIII=12 — B 3 — 2 C 2 indicates that the case XIII is a cubic surface, the 

 class whereof is 12— 7, = 5, the reduction arising from a biplanar node, B xi 

 reducing the class by 3, and from 2 conic nodes, C 2 , each reducing the clask 

 by 2. 



