ON THE SPECIAL PROBLEMS OF DYNAMICS. 233 
those of Oz, to Ow, Oy, Oz respectively are given functions of 0, ¢, c (the values 
of a’, 8", y' depending upon 6, ¢ only), we have 
pdt=sin ¢ sin ¢ d#+cos t dd, 
gdt=cos c sin ¢ d@—sin ¢ d¢, 
rdt=dc-+cos dé. 
175, A set of integrals is 
Apa +Bqg6 +Cry =k cos X, 
Apa’ + Bgh' +Cry' =k cos p, 
Apa" + Bq6"'+Cry'"=k cos y, 
equivalent to two independent equations, the values of \, p, v being such that 
cos*A+cos*z+cos*y=1; but the position of the axis of z may be chosen so 
that the values on the right-hand sides become 0, 0, &; the axis of z is then 
perpendicular to the invariable plane, the condition in question serving as a 
definition. And the three equations then give 
Ap=ka"', Bo—-p", Cr=ky", 
where the values of a”, 6”, y’ in fact are 
a"=sinc sing, B’=coscsing, y’=cos¢; 
we have thus c, ¢in terms of 7. And the equation rdt=dz + cos ¢@0 then leads 
to the value of d@, or @ is determined as a function of r by a quadrature. 
176. The constants of integration are h, &, 1 (the constant attached to 2), 
g (the constant attached to 6); and two constants, say a the longitude of 
the node, and y the inclination of the invariable plane in reference to an 
arbitrary plane of wy and origin « of longitudes therein. I remark in passing 
that Poisson obtains an elegant set of formule for the variations of the 
constants h, k, g, 1, «, y, not actually in the canonical form, but which may 
by a slight change be reduced to it. 
177. Legendre considers the problem of Rotation in the ‘Exercices de 
Calcul Intégral,’ t. ii. (1817), and the “Théorie des Fonctions Elliptiques,” 
t. 1. pp. 366-410 (1826). He does not employ the quantities p, q, r, but 
obtains de novo a set of differential equations of the second order involving 
the angles which determine the position of the principal axes with regard to 
the axes fixed in space: these angles are in fact (calling the plane of the 
fixed axes x, y the ecliptic) the longitude and latitude of one of the principal 
axes, and the azimuth from the meridian through such principal axis of an 
arbitrary axis fixed in the body and moveable with it. The solution is 
developed by means of the elliptic integrals. From the peculiar choice of 
variables there would, it would seem, be considerable labour in comparing the 
results with those of other writers, and there would be but little use in 
doing so. 
178. Poinsot’s ‘Théorie Nouvelle de la Rotation des Corps.””—The ‘Extrait? 
of the memoir was published in 1834, but the memoir itself was not published 
in extenso until the year 1851. The ‘ Extrait’ contains, however, not only the 
fundamental theorem of the representation of the motion of a body about a 
fixed point by means of the momental ellipsoid rolling on a fixed tangent 
plane, but also the geometrical and mechanical reasonings by which this 
theorem is demonstrated ; it establishes also the notions of the Poloid and 
Serpoloid curves ; and it contains incidentally, and without any developments, 
a very important remark as to the representation of the motion by means of 
the rolling and sliding motion of an elliptic cone. The whole theory (includ- 
ing that of the last-mentioned representation of the motion) is in the memoir 
