NOTES ON RELATIVE MOTION. a35 
resolutions are effected for BB’, CC’. The values of AD, DH, ete., are at 
once derived trom the displacements in time 6¢ of the points (1, 0, 0), (0, 1, 0), 
(0, 0, 1). The latter are, respectively, 
0 > 63, O55 
— 65, 0 ’ dae. 
03, 9, 0 ’ 
each multiplied by é¢ ; from which the values of AD, DH, etc., are obtained 
by multiplying the first set by OA, the second by OB, and the third by OC. 
Moreover, the parts HA’, etc., remain unchanged in magnitude when resolved 
along OZ, On, OZ, if infinitesimals above the first order be neglected. Thus, 
in the present case, HA = 1ét, AD =u0,%t, DH = — uO, 6t. 
‘4. If, in the previous case, the origin moves, its acceleration must 
of course be added to the expressions found in § 3. These formulas. 
may be tested by the following well-known example. Let O be on 
the earth’s surfacé in latitude A, and let Of be drawn south, On east, 
and Of vertical. Then w being the earth’s rotation and 7 its radius, 
the accelerations of O are 
—w’r cos i sin A along O£, 
—w°*r cos” i 5S, OR 
Also, 0,= -~ w cos J, 6, = 0, 9, = sin A, and 6, = 0 = 6, =a 6s, 
Hence the acceleration of m at (£, 7, £) are 
i wr cos 4 sin A—-2 wi, sin A—wE sin*A—w*f sin 2 cos A, 
4 +2 wf cosd+t 2 wé sin 4 — 0%, 
£— w*r cos*s — 2 wi) cos A — w% cos*h — w% sin 2 cos 2, 
along O£, On, Of, respectively. 
5. To measure the changes in the rotation of a rigid body with one 
point fixed, the axes moving asin § 3. Let the rotations to which 
the displacement of the body is due be at time ¢, o, = OA, wo, =OB, 
w, = OC measured respectively along O£, On, Of. Then since at 
time ¢ + d¢ these become w, + 0,dt= OA’, etc., along Of’, On’, 0%’, 
the absolute changes per unit time in the rotation are ultimately 
AA’ BB’ CC 
ie. tuna 
Resolving these, we get for the required components 
&, — o,f, + wh, along OE, ete. 
6. To measure the change in the whole absolute momentum of a rigid 
hody, one point of which is fixed at O, the axes moving as in §§ 3, 5. 
Since the absolute momentum of m in the position (£, 7, ) at time ¢ is 
m. {c (w, + 9.) —y (ws + ,)} along. 96, etc., 
