543 



carried about with the body, always touches the invariable 

 plane ; the side of contact, at any instant, being that which 

 corresponds to OI, and which therefore hes in the plane 

 passing through 01 and the axis of rotation. The angle de- 

 scribed in the invariable plane by the side of contact is the 

 sum or difference of two angles, one of which is proportional 

 to the time, and ihe other is the angle described by that side 

 in the surface of the cone. As the latter angle is measured 

 by the arc of a spherical conic, it followed, on comparing 

 this result with the integral given by Legendre in his dis- 

 cussion of the question of rotation, that the arc of a spheri- 

 cal conic represents an elliptic function of the third kind 

 with a circular parameter. 



The curve described by the point I on the surface of the 

 elHpsoid, is a spherical conic ; and it now appears in what 

 way the consideration of this mechanical question led to the 

 study of the properties of cones and spherical conies. From 

 theorems relating to centrifugal forces and principal axes of 

 rotation, I was further led to consider systems of eUipsoids 

 and hyperboloids having the foci of their principal sections 

 the same ; and then the focal curves presented themselves 

 as the limits of these surfaces. The properties of the focal 

 curves and of confocal surfaces occupied me, at intervals, in 

 the year 1832 ; but in the latter part of that year my atten- 

 tion was diverted from these subjects, and it was not until 

 1834 that I began to think of writing down and publishing 

 the results of my inquiries respecting them. In doing so, I 

 wished to be able to assign a geometrical origin to the sur- 

 faces of the second order, the theory of these surfaces being 

 intended to precede that of rotation ; and in seeking for 

 such an origin, I found the modular property. But not long 

 after (in the summer of 1834) happening to look into a French 

 scientific journal, I learned that M. Poinsot had just read 

 to the Academy of Sciences of Paris a memoir in which he 

 treated the question of rotation geometrically, by a method 



VOL. II. 2 z 



