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II. ON THE SURFACES OF THE SECOND ORDER. 



[Proceedings of the Royal Irish Academy, VOL. n. p. 446. Bead Nov. 30, 1843.] 



THERE is hardly any geometrical theory which more requires 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 Eotation 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 proper- 

 ties 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 analogous, 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 

 continuing the inquiry. I had done enough, however, in this 



* 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 published. The 

 investigations relative to Fresnel' s Wave -Surf ace will be found in the Transactions 

 of the Royal Irish Academy, VOL. xvi. p. 65 ; VOL. xvn. p. 241. See also VOL. 

 xxi. p. 32, of the same Transactions. 



