[95] 
HULER’S EQUATIONS OF MOTION. 
BY JAMES LOUDON, M.A., 
Professor of Mathematics and Natural Philosophy, University College, Toronto. 
1. A rigid body fixed at O has at time é rotations w, w, w, round 
the principal axes OA, OB, OC’: to determine the changes per unit 
time in these rotations. 
The positions OA’, OB’, OC’ of the axes at time ¢ + d¢ will be known 
from the displacements in time 6¢, due to these rotations, of the points 
A (@,, 0, 0), B(o, w,, 0), C(o, 0, #3). The components of these displace- 
ments in the directions OA, OB, OC, respectively, are evidently 
QO, w,0,0t, —w,,0t, for A; 
— 0,0, O, w,0,0t, for B; 
w,0,0t,  — ®,w,0t, pe tor’ C. 
: : d i 
The component rotations at time ¢ +0 are w,+ a &e., which 
¢ 
i 
may be represented by OA’, OB', OC’. The changes of the rotations 
in time 0¢ are therefore 4A’, BB’, CC’. Resolving these changes into 
the components (AF, FP, PA’), (BG, EQ, QB’), (CH, HR, RC"), in 
the directions of the axes at time ¢, we get (observing that /’P, PA’ 
ie 0, 0), &e., 
and neglecting infinitesimals above the first order) the following as 
the resultant changes in time 0¢: 
are the displacements in time 0¢ of the point / (w,+ 
AF +GQ+ HR = = — Wes + WW) ot = oat along OA ; 
FP + BG + RC= (aw, + sony W0,) Ot = a along OB ; 
C 
dt 
, y dw, dw, 
PA'+ QB + CH =(- a, + o0,- man Ot = dt along OC. 
The change it time are therefor any ahs, dey in the 
e changes per unit time a erefore —, oe 
directions OA, OB, OC, respectively. 
