234 NOTES ON RELATIVE MOTION. 
it follows that the whole absolute momentum at that time is 
2 (w, + 9,) —y (w, + 95) along Oé, 
ax (ws + 95) — z (w, + 9,) along Oy, 
y (% + 4,) —x (w, + %) along Of, 
each multiplied by J, where (x, y, 2) is the position of the centre of 
inertia. Calling these components yp, = OA, », = OB, »p, = OC, 
respectively, it follows that at time ¢-+ d¢ they become yp, + , ¢ 
= OA’ along OF, », + y,0t = OB’ along O7/, ps + p,0t = OC’ 
along Of’. The changes in the whole momentum per unit time are, 
AA’ BB’ CC’ 
mar? ar’ whose components are 
fy — py + Hy along Qe, 
ty — ve, + 5; along On, 
fg — Hy, + po8, along OF. 
therefore, 
Since « = ZW, — yw;, etc., these expressions become, on reduction, 
M times 
2 (Gy + 4,)—y (s+ 4) 0, { (or 1) © (2+ 9) Yo + (ws +4,) ZF 
+ (w+ 4) (%)0-+ 9y 4-652) —@ {(or +9)? ++ (w+ 82)? + (w+ 6) } 
for the first, with similar values for the other two. 
7. To measure the changes in the whole absolute moment of 
momentum under the same circumstances as in § 6. Since the 
absolute moment of m’s momentum at time ¢ is m times 
(or +) GG? + 2) — (on + %) 9 — (ws + 9) along 0, 
with corresponding components along Oy, Of, it follows that the 
components of the whole moment of momentum at that time are 
A (w; + 9,) — 7 (@2 -F i) — PB (wv; ++ 9) along 08, 
— 7 (w+ 6) + B (w: + 9) — @ (ws + 9) along On, 
— B (w,— 6) — @ (we. + 84) + C (ws + 4;) along OS, 
where = 3m (7? + ©), = 2m7é, ete. 
Let these components be called », == OA, », = OB, »,; = OV, respec- 
tively. Then at time ¢ + 0¢ they become », + 1,0¢ = OA’ along O08, 
¥, + y6t = OB’ along Oy’, and vz + v,0t = OC’ along Of’. Hence 
the changes of the moment of momentum per unit time are 
AA’ BB’ CC 
Oe OE) ER, 
whose components are 
¥,— v5 + 739, along 06, 
¥y— v6, + 493 along On, 
¥3— 119, + 9, along 0%, 
