798 



problem by Lagrange, refers more particularly to the memoir of 

 Poinsot, in which the motion of a body round a fixed point, and free 

 from the action of accelerating forces, is reduced to the motion of 

 a certain ellipsoid whose centre is fixed, and which rolls without 

 sliding on a plane fixed in space ; and likewise to the researches 

 of MaccuUagh, in which, by adopting an ellipsoid the reciprocal of 

 that chosen by Poinsot, he deduced those results which long before 

 had been arrived at by the more operose methods of Euler and La- 

 grange ; observing, however, that it is to Legendre that we are in- 

 debted for the happy conception of substituting, as a means of inves- 

 tigation, an ideal ellipsoid having certain relations with the actually 

 revolving body. He then states, that several years ago he was led 

 to somewhat similar views, from remarking the identity which exists 

 between the formulas for finding the position of the principal axes of 

 a body and those for determining the symmetrical diameters of an 

 ellipsoid ; and further observing that the expression for the per- 

 pendicular from the centre on a tangent plane to an ellipsoid, in 

 terms of the cosines of the angles w^hich it makes with the axes, is 

 precisely the same in form as that which gives the value of the mo- 

 ment of inertia round a line passing through the origin. Guided by 

 this analogy, he was led to assume an ellipsoid the squares of whose 

 axes should be directly proportional to the moments of inertia round 

 the coinciding principal axes of the bodj'. This is also the ellipsoid 

 chosen by MaccuUagh. Although it may at first sight appear of 

 little importance which of the ellipsoids — the inverse of Poinsot, or 

 the direct of MaccuUagh and the author — is chosen as the geome- 

 trical substitute for the revolving body, it is by no means a matter 

 of indifference w^hen 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 

 surfaces 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 of the same order, when the 

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

 the same equilateral hyperbola. 



