476 Proceedings of the Royal Irish Academy. 



wliose polar planes touch the second cubic tangent cone which can be 

 drawn from, the node to the Hessian, and whose curve of contact is the 

 curve (10). 



It has just been seen that the tangent plane to Cat T' is the plane 



and since the node 0, 0, 0, 1, - 1 is the correspondent of every point 

 on the line v - 0, w = 0, it follows that PP' V, the tangent plane at 

 T, is the plane joining the line PP' to the node, and therefore from 

 the node can be drawn to C two tangent cones whose curves of contact 

 are plane curves, the Cayleyans of the sections of the Hessian by 



V ^d ± IV ^e = 0. 



In addition, through it, pass the six planes 



^ J« = ± yJ^ = ± Jc, 



each of which touches it along a cubic curve. (Note, Art. 15.) 



The further discussion of the surface is reserved for another paper. 



