504 



was originally given by Chasles, but the properties themselves 

 to which I allude, have been overlooked both by him, and by 

 Cauchy, who has since discussed analytically the generation 

 in question in his well-known Report on M. Arayot's Memoir^ 



" Similar considerations will apply to surfaces of the second 

 degree. 



" We assume the properties of the sphere as known. Then, 



" 1. From these the properties of the wora-wzorfM/fl!/* surfaces 

 of revolution are deduced by means of a sphero-polar transfor- 

 mation. This conception has been completely elaborated by 

 M. Chasles, in the Memoirs of the Royal Academy of Brussels. 

 (Nouv. Mem. tom. v.) 



" 2. Let us now take a non-modular surface of revolution, 

 and make its centre the origin of a second transformation ; and 

 from its properties, discovered in the way just mentioned, we 

 shall infer a great number of the properties of the modular 

 surfaces of revolution, especially those belonging to two planes 

 connected with them by remarkable relations. This idea also 

 M. Chasles has suggested, though he has not developed the 

 results of it to any considerable extent. (See Liouville's 

 Journal de Math. tom. i. p. I87.)t 



" 3. If, instead of placing the origin of transformation at 

 the centre of the non-modular surface, we assume it arbitrarily 

 in space, and then operate on the surface as before, we shall 

 light on the umbilicar method, and we shall be enabled to de- 

 duce, by a uniform process, all the properties of the directive 

 planes and their poles, which arise out of that method. In 

 this way we are led to the theorems which I stated to the Aca- 

 demy on the last night of meeting, as well as to many others 

 of a similar kind. 



" 4. It is natural now to inquire what results we should 

 obtain by applying the sphero-polar transformation to the focal 



* See also the Memoire de Geometrie, appended to the Apcrgu Historiqac. 



