334 NOTES ON RELATIVE MOTION. 



it follows that tiie whole absolute momentum at that time is 



^'"2 + Q.) — y K + O3) iilong (9i;-, 



^ {<^3 + ^y) — z («! + ^i) along Or], 



y ("'1 + °i) — « {^-i + O2) along Of, 

 each multiplied by M, where (x, y, z) is the position of the centre of 

 inertia. Calling these components 11^ = OA, fi,^ = OB, //j = OC, 

 respectively, it follows that at time t -f- dt they become /x^ -f ^ 't 

 = ai' along 01', A2 + A'2<5^ = OB' along Ojy', ^Ug -)- ^^g^f =0(7' 

 along Of'. The changes in the whole momentum per unit time are, 



^1 » AA' BB' CO' , 



tnereiore, -r— > -—-, —, whose components are 



/fi — l^2^i + IJ-3^2 along 0?, 



i«2 — ^3^1 + A^i^s along Ot), 



^ — ^1^2 + l^-Pi ^-loiig ^^• 

 Since x = 2^2 — y^io ©tc, these expressions become, on reduction, 

 M times 

 ~ (or,. + 0,)~y (0,3+ 0;)+«.i {K+0i)^ + K'+02)y2 + K + ^3)4 



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 t is m times 



(<^i + ^1) ir + C==) — (wo + 0,) ?5j — (w, + 63) CI along 01, 

 with corresponding components along Orj, OZ, it follows that the 

 components of the whole moment of momentum at that time are 

 A (co, + 0:) — r ((-2 + ^2) — /3 (w3 + O3) along 04, 



— r (w, + Gi) 4- ^ ('^2 + O2) — « (^.^3 + ^3) along Or}, 



— /3 (wi — Oi) — a (w., + Oo) + 0(ws + O3) along OC, 

 where ^ = 2w (tj" -f C"), a = I'/nijC, etc. 



Let these components be called I'l = OA, v., = OB, 1^3 = 0, respec- 

 tively. Then at time i -\- 3t they become v^ -f v'^dt = OA' along 0?', 

 y.. -f v^dt = OB' along 0)?', and y^ + j'3^/ = 00' along OC. Hence 

 the changes of the moment of momentum per unit time are 



"St ' St' St' 

 whose components are 



J'l — ''2^3 + ''3°2 along 01, 

 Vi — »'3^i + »'i®3 along Or), 

 »'3 — viOj -f v-^Oi along OC, 



