On the Surfaces of the Second Order. 261 



and other parts of the theory, to open new views respecting it ; 

 and the results at which I had arrived seemed so fitted for in- 

 struction, that, when I was appointed Professor of Mathematics 

 in the University, I made them the subject of the first lectures 

 which I gave in that capacity, in the beginning of the year 1836. 

 Next year the heads of these lectures were communicated to this 

 Academy, in a Paper of which a very short abstract appeared 

 in the Proceedings* The subject soon became a favourite one 

 among the more advanced students in the University, who are, 

 for the most part, excellent geometers, and in the present Article 

 very little will be found which is not well known amongst them ; 

 very little, indeed, which was not communicated to the Academy 

 on the occasion just mentioned, or which may not be gathered, 

 in the shape of detached questions, out of the Examination 

 Papers published in the University Calendar. But as nothing 

 has yet been published on the subject in a connected form, 

 except the brief notice in the Proceedings of the Academy, and 

 as mathematicians in other countries attach some importance to 

 researches of this kind, and appear to be in quest of certain 

 principles which are familiar to us here, it seems proper to col- 

 lect together the chief results that have already been obtained, 

 in order that persons wishing to pursue these speculations may 

 be better able to judge where their inquiries should begin, and 

 in what direction further progress is most likely to be made. 



PART I. GENERATION OF SURFACES OF THE SECOND ORDER. 



1. The different species of surfaces of the second order are 

 obtained, as is usually shown in elementary treatises, by the 

 discussion of the general equation of the second degree among 

 three co-ordinates ; but it is necessary that we should also be 

 able to derive these surfaces from a common geometrical origin, 

 if we would bring them completely within the grasp of geo- 

 metry. Now as the different conic sections may (with the 



* Proceedings of the Royal Irish Academy, VOL. i. p. 89. 



