Mr. A. Cayley on the Theory of Cubic Surfaces. 493 



for the Anthracotheria, Tapirs, and Mastodons of the Molasse 

 to yield their place to Elephants, and to Ruminants which appear 

 to be the source of our existing Bovine races. The idea that 

 depressions took place along the lines of the crevasses of the val- 

 leys after the latter had been covered by a new soil need not 

 astonish us. Great spaces at the bottom of the fractures may 

 have remained unfilled; narrowings of the rock, or the large 

 size of the first blocks engulfed, may have arrested the filling 

 up of the valley ; but subsequently an increase in the weight of 

 this temporary roof, or perhaps the addition of that of the great 

 glaciers of the diluvial period, may have caused the obstacle to 

 yield, and the soil which it supported to fall in. 



LXXIV. Note on the Theory of Cubic Surfaces. 

 By A. Cayley, Esq* 



THE equation 

 AX 3 -f-BY 3 + 6CRST = 0, 



where X + Y + R+S + T = 0, represents a cubic surface of a spe- 

 cial form, viz. each of the planes R = 0, S = 0, T = is a triple 

 tangent plane meeting the surface in three lines ivhichpass through 

 a point t; and, moreover, the three planes AX 3 -fBY 3 = are 

 triple tangent planes intersecting in a line. It is worth noti- 

 cing that the equation of the surface may also be written 



aaP + by 3 + c(u 3 -f v 3 + iv 3 ) = 0, 



where a?-\-y + u + v + w = 0. In fact, the coordinates satisfying 

 the foregoing linear equations respectively, we have to show that 

 the equation 



AX 3 + BY 3 + 6CRST = ax 3 + by 3 + c{u 3 + v 3 + w 3 ) 



* Communicated by the Author. 



t The tangent plane of a surface intersects the surface in a curve having 

 at the point of contact a double point, and in like maimer a triple tangent 

 plane intersects the surface in a curve with three double points, viz. each 

 point of contact is a double point ; there is not in general any triple tan- 

 gent plane such that the three points of contact come together; or, what 

 is the same thing, there is not in general any tangent plane intersecting the 

 surface in a curve having at the point of contact a triple point. A surface 

 may, however, have the kind of singularity just referred to, viz. a tangent 

 plane intersecting the surface in a curve having at the point of contact a 

 triple point ; such tangent plane may be termed a ' tritom ' tangent plane, 

 and its point of contact a ' tritom ' point : for a cubic surface the intersec- 

 tion by a tritom tangent plane is of course a system of three lines meeting 

 in the tritom point. The tritom singularity is sibi-reciprocal ; it is, I 

 think, a singularity which should be considered in the theory of reciprocal 

 surfaces. 



