[231] 
NOTES ON RELATIVE MOTION. 
BY JAMES LOUDON, 
University College, Toronto. 
1. Motion of a point in a plane. 
At time ¢ let the moving axes be O&, Oy, and P a point (€, 7) in 
their plane. At time ¢-+ 0¢ let these axes coincide with O&', On’, 
and P with P’; then the & and 7 components of the displacement 
PP’ are — w7St, w&dt, respectively, if w is the rate at which the axes 
turn round O£. Let a moving point be at P at time ¢, and at Q at 
time ¢ ++ of, the co-ordinates of Q referred to 0’, On’ being € + £0t, 
7 ++ 70¢; then the absolute velocity of the moving point is ultimately 
PQ PP’ PQ 
ey 
w&, respectively. 
), the € and 7 components of which are £ — wy, i + 
Putting € — wy =u = OA, and 9 + wF = v = OB, the component 
velocities at time ¢ + d¢ become wu + dt = OA’ along O&', and v + 
védt = OB' along Oy’. Hence the absolute acceleration ultimately — 
(or =r), the components of which are 
“ — vw = & — 2 of — 46 — w’E along O86, 
» + ww = 7 + 2 wF + Es —w*y along On. 
2. Motion of a rigid body round a fixed axis O£, the axes O&, On 
being fixed in the body. 
At time ¢ the whole momentum is— Mwy = OA along O&, and 
Mw& = OB along On, where &, 7 are co-ordinates of the centre of 
inertia. At time ¢ + d¢ the momentum is — My (w + dt) = OA’ 
along O&’, and ME (w + odt) = OB' along O07. The changes of 
momentum per unit time are, therefore, ultimately = = whose 
components are 
— Myo — Mw*é along O&, 
Méo — Mwy along On. 
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