Kmjal Society. 313 



of indifference wlien we come to treat of the properties of the in- 

 tegrals which determine the motion. Generally those integrals de- 

 pend on the properties of those curves of double flexure in which 

 cones of the second degree are generally intersected by concentric 

 spheres ; and it so happens that the direct ellipsoid of moments is 

 intersected by a concentric sphere in one of these curves. By means 

 of the properties of these curves a complete solution may be ob- 

 tained even in the most general cases, to which only an approxi- 

 mation has hitherto been made. 



In the first section of the paper, the author establishes such pro- 

 perties as he has subsequently occasion to refer to, of cones of the 

 second degree, and of the curves of double curvature in which these 

 s!irfaces may be intersected by concentric spheres, some of which 

 he believes will not be found in any published treatise on the sub- 

 ject. He considers that he has been so fortunate as to be the first 

 to obtain the true representative curve of elliptic functions of the 

 first order. It is shown that any spherical conic section, the tan- 

 gents of whose principal semiarcs are the ordinates of an equilateral 

 hyperbola whose transverse semiaxis is 1, may be rectified by an 

 elliptic function of the first order; and the quadrature of such a 

 curve may be effected by a function oi" the same order, when the 

 cotangents of the halves of the principal arcs are the ordinates of 

 the same equilateral hyperbola. 



This particular species of spherical ellipse the author has called 

 the " Parabolic Ellipse," because, as is shown in the course of the 

 investigation, it is the gnomonic projection, on the surface of a sphere, 

 of the common parabola whose plane touches the sphere at the focus. 

 As in this species of spherical ellipse either the focus or the centre 

 may be taken as the origin of the spherical radii vectores, in effect- 

 ing the process of rectification, we are unexpectedly presented with 

 Lagrange's scale of modular transformations, as also with the other 

 equally well-known theorem by which the successive amplitudes are 

 connected. Among other peculiar properties of the spherical para- 

 bolic ellipse established in this paper, it is shown that the portion of 

 a great circle touching the curve, and intercepted between the per- 

 pendicular arcs on it from the foci, is always equal to a quadrant. 



In the second and following sections, the author proceeds to dis- 

 cuss the problem which is the immediate subject of the paper. 

 Having established the ordinary equations of motion, he shows that, 

 if the direct ellipsoid of moments be constructed, the motion of a 

 rigid body acted on solely by primitive impulses may be represented 

 by this ellipsoid moving round its centre, in such a manner that its 

 surface shall always pass through a point fixed in space. This point, 

 so fixed, is the extremity of the axis of the plane of the impressed 

 couple, or of the plane known as the invariable plane of the motion. 



But a still clearer idea of the motion of such a body is presented 

 in the subsequent investigations, it being there shown, that the most 

 general motion of a body round a fixed point may be represented by 

 a cone rolling with a certain variable velocity on a plane whose axis 

 is fixed, while this plane revolves about its own axis with a certain 



