799 



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 

 ma}^ be taken as the origin of the spherical radii vectores, in eflPect- 

 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, lie 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 

 uniform velocity. This cone may always be determined. For the 

 circular sections of the invariable cone coincide with the circular 

 sections of the ellipsoid of moments ; whence the cyclic axes of 

 the ellipsoid, or the diameters perpendicular to the planes of these 

 sections, will be the focal lines of the supplemental cone ; and as 

 the invariable plane is always a tangent plane to this cone, we have 

 sufficient elements given to determine it. 



From these considerations it appears that we may dispense alto- 

 gether with the ellipsoid of moments, and say that if two right lines 

 be drawn through the fixed point of the body in the plane of the 

 greatest and least moments of inertia, making angles with the axis 

 of greatest moment, the cosines of which shall be equal to the 

 square root of the expression 



L(M-N) 

 M(L-N)' 



(L, M, N being the symmetrical moments of inertia round the prin- 

 cipal axes) and a cone be conceived having those lines as focals, and 

 touching moreover the invariable plane, the motion of the body will 

 consist in the rotation of this cone on the invariable plane with a 

 variable velocity, while the plane revolves round its own axis with 

 an uniform velocity. 



Although -it is very satisfactory, the author remarks, in this wav 



