244, REPORT—1862. 
dition moment=0, will not of necessity be in equilibrium ; such six lines are 
said to be in involution, and the geometrical theory is a very extensive and 
interesting one. 
Miscellaneous Problems. Article Nos. 216 to 223. 
216. As under the foregoing head, “‘ Rotation round a fixed point,” I have 
considered only the case of a body not acted upon by any forces, the case 
where the body is acted upon by any forces comes under the present head. 
The forces, whatever they are, may be considered as disturbing forces, and 
the problem be treated by the method of the variation of the elements ; this 
is at any rate a separate part of the theory of rotation round a fixed point, 
and I have found it convenient to include it under the present head; the 
only case in which the forces have been treated as principal ones, seems to be 
that of a heavy body (a solid of revolution) rotating about a point not its 
centre of gravity. The case of a body suspended by a thread or resting on a 
plane comes under the present head, as also would (in some at least of the 
questions connected with it) the gyroscope. But none of these questions are 
here considered in any detail. 
Rotation round a fixed point—Variation of the elements. 
217. The forces acting on the body are treated as disturbing forces. 
Formule for the variations of the elements were first obtained by Poisson 
in the memoir “ Sur la Variation des Constantes Arbitraires &c.” (1809). The 
variations are expressed in terms of the differential coefficients of the disturb- 
ing function in regard to the elements, and, as the author remarks, they are 
very similar in their form to, and can be rendered identical with, those for 
the variations of the elements in the theory of elliptic motion. 
218. Cayley, “On the Rotation &c.” (1846).—The latter part of the paper 
relates to the variations of the elements therein made use of, which are 
different from the ordinary ones. 
219. Richelot, “Eine neue Lésung &e.” (1851).—The form in which the 
integrals are obtained by means of a function V, satisfying a partial differen- 
tial equation, leads at once to a canonical system for the variations of the 
elements; these formule are referred to in the introduction to the memoir, 
but they are not afterwards considered. 
220. Cayley, “ On the Rotation of a Solid Body” (1860).—The elements are 
those ordinarily made use of, with only a slight variation occasioned by the 
employment of the “ departure” of the node. The course of the investigation 
consists in obtaining the variations in terms of the differential coefficients of 
the disturbing function in regard to the coordinates (formule which were 
thought interesting for their own sake), and in deducing therefrom those in 
terms of the differential coefficients in terms of the elements. 
Other cases of the motion of a solid body. 
221. In regard to a heavy solid of revolution rotating about a fixed point 
not its centre of gravity, we have 
Poisson, “‘ Mémoire sur un cas particulier &c.” (1831), and the elaborate 
memoir 
Lottner, “ Reduction der Bewegung &c.”’ (1855), where the solution is 
worked out by means of the Elliptic and Jacobian functions, 
222. As regards a heavy solid suspended by a string, 
Pagani, “‘ Mémoire sur l’équilibre Ke.” (1839). 
223, As regards the motion of a body resting on a fixed plane, 
