ON THE SPECIAL PROBLEMS OF DYNAMICS, 243 
of an infinitesimal motion, each point of it along the line on which the same 
is situate. 
212. The entire theory, as well of the infinitesimal as of the finite motions 
of a solid body, is carefully and successfully treated in Rodrigues’ memoir 
“ Des lois géométriques &c.” (1840). It may be remarked that for the purpose 
of compounding together any rotations and translations, each rotation may be 
resolved into a rotation about a parallel axis and a couple of rotations, that 
is, a translation; the rotations are thus converted into rotations about axes 
through one and the same point, and these give rise to a single resultant 
rotation given as to its magnitude and the direction of the axis, but not as to 
the position of the axis (which is an arbitrary line in the given direction) ; 
the translations are then compounded together into a single translation, and 
finally the position of the axis of rotation is so determined that the translation 
shall be in the direction of this axis; the entire system is thus compounded 
(in accordance with Chasles’ theorem) into a rotation and a translation in the 
direction of the axis of the rotation. The problem of the composition depends 
therefore on the composition of rotations about axes through one and the 
same point; that is, upon Euler’s and Lexell’s theory of the resultant axis. 
But, as already noticed, the analytical theory of the resultant axis was per- 
fected by Rodrigues in the present memoir (see ante, ‘ Transformation of Co- 
ordinates,’ Nos. 139-141, as to this memoir and the quaternion representation 
of the formulee contained in it). 
213. It was remarked in Poinsot’s memoir of 1834 that every continuous 
motion of a solid body about a fixed point is the motion of a cone fixed in 
the body rolling upon another cone fixed in space. The corresponding theorem 
for the motion of a solid body in space is given 
Cayley, “On the Geometrical Representation &c.” (1846), viz. premising that 
a skew surface is said to be “‘ deformed” if, considering the elements between 
consecutive generating lines as rigid, these elements be made in any manner 
to turn round and slide along the successive generating lines :—and that two 
skew surfaces can be made to roll and slide one upon the other, only if the 
one is a deformation of the other—and that then the rolling and sliding 
motions are perfectly determined—and that such a motion may be said to be 
a “gliding” motion: the theorem is that any motion whatever of a solid body 
in space may be represented as the gliding motion of one skew surface upon 
another skew surface of which it is the deformation. 
214. The same paper contains the enunciation and analytical proof of the 
following theorem supplementary to that of Poinsot just referred to, viz. 
that when the motion of a solid body round a fixed point is represented as 
the rolling motion of one cone on another, then “the angular velocity round 
the line of contact (the instantaneous axis) is to the angular velocity of this 
line as the difference of the curvatures of the two cones at any point in this 
line is to the reciprocal of the distance of the. point from the vertex.” 
215. There are a great number of theorems relating to the composition of 
forces and force-couples, which consequently relate also to infinitesimal rota- 
tions and translations. See, for instance, Chasles, “ Théorémes généraux ce.” 
(1847), Mobius, “ Lehrbuch der Statik” (1837), Steichen’s Memoirs of 1853 
and 1854, &c. Arising out of some theorems of Mobius in the “ Statik,” we 
have Sylvester's theory of the involution of six lines: viz. six lines (given in 
position) may be such that properly selected forces along them (or if we 
please, properly selected infinitesimal rotations round them) will counter- 
balance each other; or, what is the same thing, the six lines may be such 
that a system of forces, although satisfying for each of the six lines the con- 
R2 
