REPORT ON THE PROGRESS OF THE SOLUTION OF 
580 
[298 
as the complete integrals of either problem, the last three of them being the final 
integrals. 
And it is added that if in either problem we have H + Q instead of H, the expressions 
for the variations of the elements assume the canonical forms ^ = — ^rr , = ~r^, &c. 
at db at da 
The solution is not further developed as regards the rotation problem, but it is 
so (§ 67) as regards the other problem. 
207. It must, I think, be considered that a comprehensive memoir on the Problem 
of Rotation, embracing and incorporating all that has been done on the subject, is 
greatly needed. 
Kinematics of a solid body. Article Nos. 208 to 215. 
208. The general theorem in regard to the infinitesimal motions (rotations and 
translations) of a solid body is that these are compounded and resolved in the same 
way as if they were single forces and couples respectively. Thus any infinitesimal 
rotations and translations are resolvable into a rotation and a translation ; the rotation 
is given as to its magnitude and as to the direction of its axis, but not as to the 
position of the axis (which may be any line in the given direction) : the magnitude 
and direction of the translation depend on the assumed position of the axis of rotation ; 
in particular this may be taken so that the translation shall be in the direction of 
the axis of rotation ; and the magnitude of the rotation is then a minimum. I remark 
that the theorem as above stated presupposes the establishment of the theory of couples 
(of forces) which was first accomplished by Poinsot in his ‘ Elémens de Statique,' 1st edit. 
1804; it must have been, the whole or nearly the whole of it, familiar to Chasles at 
the date of his paper of 1830 next referred to (see also Note XXXIV. of the Aperçu 
Historique, 1837); and it is nearly the whole of it stated in the ‘Extrait’ of Poinsot’s 
memoir on Rotation, 1834. 
209. Chasles’ paper in the Bulletin TJniv. des Sciences for 1830.—The corresponding 
theorem is here given for the finite motions (rotations and translations) of a solid 
body as follows : viz. if any finite displacement be given to a free solid body in space, 
there exists always in the body a certain indefinite line which after the displacement 
remains in its original situation. The theorem is deduced from a more general one 
relating to two similar bodies. It may be otherwise stated thus : viz., any motions may 
be represented by a translation and a rotation (the order of the two being indifferent) ; 
the rotation is given as regards its magnitude and the direction of its axis, but not 
as to the position of the axis (which may be any line in the given direction) ; the 
magnitude and direction of the translation depend on the assumed position of the axis 
of rotation ; in particular this may be taken so that the translation shall be in the 
direction of the axis of rotation ; the magnitude of the translation is then a minimum. 
It may be noticed that a translation may be represented as a couple of rotations ; 
that is, two equal and opposite rotations about parallel axes.