82 Journal of the Mitchku, Society. [June 



been done upon the subject would enlarge the present paper 

 into a book. It was found impossible to cover even the 

 geometrical phases of the problem, in their extension in par- 

 ticular to the cognate problem of the forty-five triple tangent 

 planes, although the two subjects go hand in hand. In this 

 memoir, however, is given a general survey of the problem of 

 the twenty-seven lines, from the geometric standpoint, with 

 special attention to salient features, i. e., the concept of 

 trihedral pairs, the configuration of the double-six, the solu- 

 tion of the problem of constructing models of a double-six 

 and of the configurations of the lines upon the twenty-one 

 types of the cubic surface, the derivation of the Pascaliau 

 configuration from that of the lines upon the cubic surface 

 with one conical point, and certain allied problems. 



In §§ 1-4 are given certain preliminary theorems concerning 

 the existence and number of the twenty-seven lines and forty- 

 five planes for the general cubic surface, and upon the -first 

 notation employed. In §§ 5, 6 and 7 are given an account of 

 Schlafli's notation, a history of the double-six theorem and an 

 analytic proof of it, independent of cubic surfaces; in §8 follow 

 certain interesting results on the anharmonic ratios of the con- 

 figurations. In §9 appear two conditions that five lines lie upon 

 a cubic surface and in §10 is the description of the formation, 

 and the tabulation of the thirty-six double-sixes. In §11 

 occur certain auxiliary theorems for special features of the 

 general configuration of the twenty-seven lines. 



In §12 are given the definition and number of trihedral 

 pairs, and in §13 the actual formation of the tables of the 120 

 forms. In §14 these are grouped together in such a way (sets 

 of three) as to determine in forty ways all the twenty-seven 

 lines. 



In §16 is given a discussion of a special form of the general 

 equation of the cubic surface and the determination of the 

 equations of the forty-five triple tangent planes. 



In §§18 and 19 the methods for the construction of a model 



