24 
[388 
388. 
NOTE ON THE COMPOSITION OF INFINITESIMAL ROTATIONS. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. vm. (1867), 
PP- 7-10.] 
The following is a solution of a question proposed by me in the last Smith’s 
Prize Examination : 
“Show that infinitesimal rotations impressed upon a solid body may be compounded 
together according to the rules for the composition of forces.” 
Definition. The “six coordinates” of a line passing through the point (x 0 , y 0 , z 0 ), 
and inclined at angles (a, /3, 7), to the axes, are 
a = cos a, f = y 0 cos 7 — z 0 cos /3, 
h = cos /3, y = 2 0 cos a — x 0 cos 7, 
c = cos 7, h = x 0 cos yS — 2/0 cos a.. 
I use, throughout, the term rotation to denote an infinitesimal rotation; this 
being so, 
Lemma 1. A rotation &> round the line (a, b, c, f g, h), produces in the point 
{x, y, z), rigidly connected with the line, the displacements 
8x = (o( . cy — bz+f), 
By = co (— cx . +az + g), 
Sz = a) ( bx — ay . + h). 
Lemma 2. Considering in a solid body the point (x, y, z), situate in the line 
(a, b, c, f, g, h), then for any infinitesimal motion of the solid body, the displacement 
of the point in the direction of the line is 
= al + bm 4- cn +fp+gq + hr,