2352 NOTES ON RELATIVE MOTION. 
At time ¢ the whole moment of momentum is (employing OA, OB 
in a new sense) 
— Bw = OA along 0, 
— aw = OB along 07, 
Cw... . along OZ, 
where a= 2m7il, C= 2m (+ 7’), etc. 
At time ¢ + 6¢ the moment of momentum becomes 
— 6 (wv + ot) = OA’ along OF, 
—a(w-+ ot) = OF along 07, etc. 
Hence the changes per unit time of moment of momentum are ulti- 
mately a “= Co, the components of which are — (36 + aw? along 
0, — a@ — Bw? along On, and Cw along OZ. 
These, it will be observed, are of the same form as when the axes 
are fixed in space. 
3. To measure the absolute velocity and acceleration of a point 
referred to axes moving in space round 0. 
Let the motion of the axes be due to rotations @,, 0,, 0, measured 
along themselves. Then, proceeding as in § 1, the displacements of 
a point P (é, y, €) due to these rotations are (£0,— 76;) ot along O€, 
(£0, — £0,) dt along Oy, and (40, — £6.) dt along Of. These added to 
the relative displacements (Et, Ot, 6t) of the moving point give 
the absolute displacements. Hence the components of the absolute 
velocity are 
u=O0A=£+ 66,—70, along O£, 
© = OB =i, + £6, — £0, along On, 
w= OC =£ + 70,— £6, along O£. 
Again, let the velocities at time ¢ + dt be OA’ = u + wot along 
AA’ BB CC’ 
Oé', etc.; then the absolute accelerations are ultimately Fr? ee? 
whose components are 
wu — v0, + wy along OE, 
v—wh, + ud, along 07, 
w — U0, + vO, along OF. 
These become, on reduction, 
E—2 O53 -+ 2 0,6 + £0,—70,— (8; + 67 + 03) & + (60, + 99; + £02) 6; ° 
along OG, ete. 
Nors.—These resolutions are most readily effected as follows: AA’ is 
equivalent to AD along On, DH along 02, and HA’ along O€ ; and similar 
