242 REPORT—1 862. 
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 resolvible 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 Apercu 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 Uniy. 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 inde- 
finite 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 trans- 
lation 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. 
210. It is part’of the general theorem that any number of rotations about 
axes passing through one and the same point may be compounded into a 
single rotation about an axis through that point ;. this is, in fact, the theory 
of the “ Resultant Axis ” déyeloped in Euler’s and Lexell’s memoirs of 1775. 
211. The following properties are also given, viz., considering two similar 
solid bodies (or in particular any two positions of a solid body) and joining 
the corresponding points, the lines which pass through one and the same 
point form a cone of the second order; and the points of either body form 
on this cone a curve of the third order (skew cubic). And, reciprocally, the 
lines, intersections of corresponding planes, which lie in one and the same 
plane envelope a conic, and such planes of either body envelope a developable 
surface, which is such that any one of these planes meets it in a conic [or, 
ee is the same thing, the planes envelope a developable surface of the fourth 
order}. 
And also, given in space two equal bodies situate in any manner in respect 
to each other, then joining the points of the first body to the homologous 
points of the second body, the middle points of these lines form a body capable 
: 
: 
