447 



it; and the results at which I had arrived seemed so fitted 

 for instruction, 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 be- 

 ginning 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 ad- 

 vanced 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 Aca- 

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

 gathered, in the shape of detached questions, out of the 

 Examination-Papers published yearly in the University Ca- 

 lendar. But as nothing has yet been published on the sub- 

 ject in a connected form, except the brief notice in the Pro- 

 ceedings 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 collect 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 de- 

 gree among three coordinates ; 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 



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



