26 
NOTE ON THE COMPOSITION OF INFINITESIMAL ROTATIONS. 
[388 
and the expressions which enter into this formula denote as follows; viz. if through 
the point P, at right angles to the plane through P and the axis of rotation, we 
draw a line PQ, = perpendicular distance of P from the axis of rotation, then the 
coordinates of Q referred to P as origin are 
— y cos 7 + z cos /3, 
X cos 7 . — z cos a, 
— X cos ß + y cos a . , 
respectively. Hence the foregoing quantities each multiplied by co are the displacements 
of the point P in the directions of the axes, produced by the rotation co. Suppose 
that the axis of rotation (instead of passing through the origin) passes through the 
point (x 0 , y 0 , z 0 ) ; the only difference is that we must in the formulae write 
(x — x 0 , y — y 0 , z — z 0 ) in place of (x, y, z): and attending to the significations of the 
six coordinates (a, b, c, f, g, h) it thus appears that the displacements produced by the 
rotation are equal to co into the expressions 
. -cy+ bz +/, 
respectively. 
cx . — az + g, 
— bx + ay . + h, 
Demonstration of Lemma 2. 
For any infinitesimal motion whatever of a solid body, the displacements of the 
point (x, y, z) in the directions of the axes are 
Bx = l . — ry + qz, 
By — m + rx . —pz, 
Bz = n — qx +py . , 
and hence the displacement in the direction of the line (a, ¡3, 7), is 
Bx cos a + By cos /3 +• Bz cos 7, 
which, attending to the signification of the six coordinates (a, b, c, f, g, h), is 
= al + bm + cn +fp + gq + hr, 
which is the required expression. 
It is proper to remark that the last-mentioned expressions of (Bx, By, Bz) are in 
fact the displacements produced by a translation and a rotation. If we assume that 
every infinitesimal motion of a solid body can be resolved into a translation and a 
rotation, then, since a translation can be produced by two rotations, every infinitesimal 
motion of a solid body can be resolved into rotations alone, and the foregoing expressions 
for the displacements produced by a rotation, combining any number of them and 
writing (Icoa, Zcob, Icoc, Icof 2cog, 2coh) — (-p, — q, —r, l, m, w) respectively, lead to the 
expressions for the displacements Bx, By, Bz produced by the infinitesimal motion of the 
solid body.