ON THE SPECIAL PROBLEMS OF DYNAMICS. 237 
manageable form; the reason of this is the presence in the expression for 0 of 
the non-periodic term —n'u. It will presently be seen how this difficulty was 
got over by Jacobi. 
190. Briot’s paper of 1842 contains an analytical demonstration of some 
of the theorems given in the ‘ Extrait’ of Poinsot’s memoir of 1834. 
191. In Maccullagh’s Lectures of 1844 (see Haughton, 1849; Maccullagh, 
1847) the problem of the rotation of a solid body is treated in a mode some= 
what similar to that of Poinsot’s; only the ellipsoid of gyration ~ nt+S=1 7 
if A, B, C=Ma’*, Mb’, Mc’) is used instead of the momental ellipsoid. Thus, 
reciprocal to the poloid curve on the momental ellipsoid we have on the 
ellipsoid of gyration a curve all the points whereof are equidistant from the 
centre ; such curve is of course the intersection of the ellipsoid by a concen- 
tric sphere, that is, it is a spherical conic; and the points of this spherical 
conic come successively to coincide with a fixed point on the invariable axis. 
This is a theorem stated in Art. VI. of Haughton’s memoir: it may be added 
that the several tangent planes of the ellipsoid of gyration at the points of the 
spherical conic as they come to coincide with the fixed point, form a cone 
reciprocal to Poinsot’s serpoloid cone. It is clear that every theorem in the 
one theory has its reciprocal in the other theory; I have not particularly 
examined as to how far the reciprocal theorems haye been stated in the two 
theories. 
192. Cayley, “ On the Motion of Rotation of a Solid Body ” (1843).—The 
object was to apply to the solution of the problem Rodrigues’ formule for the 
resultant rotation ; viz., if the principal axes, considered as originally coin- 
ciding with the axes of «, y, z, can be brought into their actual position at the 
end of the time ¢ by a rotation 6 round an axis, inclined at angles f, g, h to 
the axes of w, y, z, and if \=tan $0 cos f, »=tan 46 cos g, v=tan 36 cosh, 
then the principal axes are referred to the axes fixed in space by means of 
the quantities \, », v. And these are to be obtained from the equations 
kpdt=2( dd+ vdu—pdy), 
« gdt=2(—rvdi\+ du+ddr), 
crdt=2( pdr—ddu+ dy), 
where k=1+)°+ p°++ ’, and p, q, 7 are to be considered as given functions 
of ¢, or of other the variable selected as the independent one. But for effecting 
the integration it was found necessary to take the axes of z as the invariable 
axes. 
193. The solution by Jacobi, § 27 of the memoir “Theoria Novi Multi- 
plicatoris” (1845), is given as an application of the general theory, the author 
remarking that, as well in this question as in the problem of the two fixed 
centres, he purposely employed a somewhat inartificial analysis, in order to 
show that the principle (of the Ultimate Multiplier) would lead to the result 
without any special artifices. The principal axes are referred to the axes 
fixed in space by the ordinary three angles (here called q,, q,, q,), and the 
solution is carried so far as to give the integral equations, without any reduc- 
tion of the integrals cohtained in them to elliptic integrals. The solution is, 
howeve¥, in so far remarkable that the integrations are effected without the 
aid of the invariable plane. 
194. Cayley, “On the Rotation of a Solid Body &c.” (1846).—It appeared 
desirable to obtain the solution by means of the quantities A, HB, v, without the 
assistance of the invariable plane, and Jacobi’s discovery of the theorem of the 
