of Surfaces of the Secofid Order. 433 



cipal planes of a surface of the second order, there exist conic 

 sections, termed by him eccentric conies, confocal to the 

 sections of the surface in the principal planes; possessing 

 properties analogous to the foci of curves of the second order; 

 and cylinders perpendicular to those planes, bearing an ana- 

 logy to the directrices of the same curves;" but this and other 

 theorems of the same class, may be considered rather as the 

 limiting relations of confocal surfaces, than as analogous to 

 the well-known properties of curves of the second degree ; 

 and this view of the subject is further confirmed by the consi- 

 deration, that the theorems above alluded to fail in the very 

 case most analogous to that of the conic sections, when the 

 surface is one of revolution round the transverse axe. 



M. Chasles, indeed, in a very elegant memoir published 

 now more than ten years ago *, has given several of the ana- 

 logous theorems in the case of surfaces of revolution round 

 the transverse axe, but has not hitherto extended his re- 

 searches to the case of oblate spheroids, or to that of surfaces 

 having three unequal axes, to do which is the object of the 

 following pages. 



By a simple application of a new method, which has now r 

 been for some time published f? I have been led to the dis- 

 covery of a very extensive class of properties of surfaces of 

 the second degree, hitherto, I believe, entirely unknown ; from 

 which may be easily deduced a series of theorems, relative to 

 curves of the second order, none of which, so far as I am 

 aware, have been yet given to the public. 



The restricted limits of the present communication pre- 

 clude the possibility of giving more than an outline of the 

 theory, and the enunciations of several new theorems, the 

 demonstrations of several of which being somewhat tedious, 

 have been suppressed ; but this can cause no difficulty to any 

 moderately expert analyst. 



(1.) Let a, b, c, denote the three semiaxes in the order of 

 magnitude of a surface of the second order, which for brevity 

 may be represented by the symbol (£), and let the eccen- 

 tricities of the three principal sections of the surface (2) be 

 e, e, >j, so that 



„ a*-b* a a*-c 2 n b*-c* 





(2.) Let u denote the semidiameter of the surface passing 



\ * See the Kouveaux Memoires de V Academie TLoyale de Bruxelles, vol. v. 

 f See a short treatise by the author, " On the application of a new 

 Analytic Method to the theory of Curves and Curved Surfaces.'* Dublin, 

 Hodges and Smith, 1840. 



Phil. Mag. S. 3. Vol. 17. No. 112. Pec. 1840. 2 F 



