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II. Oa a Special Form of the General Equation of a Cubic Surface and on a 
Diagram Representing the Tiventy-seven Lines on the Surface * 
By H. M. Tayloe, M.A., Fellow of Trinity College, Cambridge. 
Communicated by h . R. Foesyth, Sc.D., F.R.S. 
Received June 13,—Read June 15, 1893. 
Revised October 30, 1893. 
The existence of straight lines on a cubic surface, the number of them, aud their 
relations to each other was first discussed in a correspondence between Salmoy and 
Cayley. ‘ 
In a paper which appeared in 1849, in vol. 4 of the ‘Cambridge and Dublin 
Mathematical Journal,’ “ On the Triple Tangent Planes of Surfaces of the Third 
Order,” Caa^ley gave a sketch of what was then known, and gave the equations of 
the forty-five planes in which the twenty-seven lines on the surface lie by tlu’ees, 
when the equation of the surface is taken in a particular form. 
In the above-mentioned paper, Cayley remarks, “ there is great difiiculty in 
conceiving the complete figure formed by the twenty-seven lines : indeed, this can 
hardly, I think, be accomplished until a more perfect notation is discovered.” 
ScHLAFLit has discovered a notation of great merit which affords a powerful 
method of dealing with the twenty-seven lines ; it is based upon the selection of 
some twelve of the lines which form a “ double six.” The author of this paper 
endeavoured to find a notation for the twenty-seven lines, which did not depend on 
any special selection among them. He hopes that the method he has adopted of 
representing by a plane diagram the intersection or non-intersection of the twenty- 
seven lines with eacli other will be found of some interest. 
Foiu’ distinct forms of the diagram are given : one will be found of more use for 
one purpose, and another for another; although each contains everything that is 
contained in the others. In fact, one is obtained from another by purely clerical 
alteration. 
The contents of this paper may be stated shortly as follows :— 
In § 1 it is shown that the equation of the general cubic surface may be thrown 
into the form 
* As originally communicated, this papier was entitled, “ On a Grapliical Rein’esentation of tbe 
Twenty-seven Lines on a Cubic Surface.” 
t ‘ Quarterly Joiumal of Matbematics,’ vol. 2, p. 113. 
22.2.94 
