446 



Professor Mac Cullagh gave an account of his researches 

 in the Theory of Surfaces of the second Order, in connexion 

 with a former communication which he had made to the Aca- 

 demy on the same subject. These researches are contained 

 in the following paper. 



On the Surfaces of the Second Order. 

 There is hardly any geometrical theory which more re- 

 quires to be studied, or which promises to reward better 

 whatever thought may be bestowed upon it, than that of 

 the surfaces of the second order. My attention was drawn 

 to it, many years ago, by the consideration of mechanical 

 and physical questions. In the dynamical problem of the 

 Rotation of a Solid Body, and in the investigation of the 

 properties of the Wave-Surface of Fresnel, I found, so long 

 since as the year 1829, that the ellipsoid could be employed 

 with very great advantage ; while the discussion of these 

 questions, but especially of the former,* suggested pro- 

 perties of the ellipsoid and its kindred surfaces which I 

 might not otherwise have perceived. In this manner I 

 was led to consider systems of confocal surfaces, and thence 

 to notice the focal curves, which I discovered to be analo- 

 gous, in the theory of the surfaces of the second order, to 

 the foci in that of the plane conic sections. That theory 

 now began to interest me on its own account, and, guided 

 by analogy, I struck out the leading properties possessed 

 by the surfaces in relation to their focal curves ; but the 

 interference of other matters prevented me from continu- 

 ing the inquiry. I had done enough, however, in this and 

 other parts of the theory, to open new views respecting 



* The Theory of Rotation, here spoken of, was completed in the year 1831 ; 

 but, from causes which need not be mentioned at present, it was not pub- 

 lished. The investigations relative to Fresnel's Wave-Surface will be found in 

 the Transactions of the Royal Irish Academy, vol. xvi. p. 65; vol. xvii. p. 241. 

 See also vol. xxi. p, 32, of the same Transactions. 



