NOTES ON RELATIVE MOTION. 23^ 



resolutions are effected for BB', CC. The values of AD, DH, etc., are at 

 once derived from the displacements in time St of the points (1, 0, 0), (0, 1, 0), 

 (0, 0, I). The latter are, respectively, 



, ^3 , 02 , 

 — e,, , 01, 



02, «i, , 

 each multiplied by St ; from which the values of A D, DH, etc. , are obtained 

 by multipljring the first set by OA, the second by OB, and the third by 00. 

 Moreover, the parts HA', etc., remain unchanged in magnitude when resolved 

 along 0^, 0)], 0Z„ if infinitesimals above the first order be neglected. Thus, 

 in the present case, HA ^=uSt, AD = uQ^U, DH=^ — uO^dt. 



i. 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 be on 

 the earth's surface in latitude A, and let 0^ be drawn south, Orj east, 

 and 0? vertical. Then w being the earth's rotation and ?• its radius, 

 tlie accelerations of are 



— w'^r cos A sin X along 0^, 



— (o^r cos^ X " 0^. 



Also, 6^= — (o cos A, 02 = 0) ^s = ^ sin A, and 9^ = = 9^ = 6^. 

 Hence the acceleration of m at (^, r], C) are 



t, — (o'^r cos X sin X — 2 (o-i] sin A — cu^^ sin^A — cu'-'^ sin A cos A, 



-/) -\- 2 wZ cos A -|- 2 w^ sin A — (o'^yj, 



C — (o'^r cos^A — 2 (ufj cos A — - co^Z cos^A — w^4 sin A cos A, 



along 0^, Ot}, OZ, respectively. 



5. To measure the changes in the rotation of a rigid body with one 

 point fixed, the axes moving as in § 3. Let the rotations to which 

 the displacement of the body is due be at time t, w^ =. OA, w^ =iOB, 

 Wg = OC measured respectively along 0^, Of], 0^. Then since at 

 time t -\- M these become w^ -f- (J\U = OA', etc., along 0|', Or]', 0^\ 

 the absolute changes per unit time in the rotation are ultimately 



AA' BB' GC 



' U' It' St' 



Resolving these, we get for the I'equired components 

 a>i — ty.2^3 -f- ^3(32 along 0^, etc. 



6. To measure the change in the whole absolute momentum of a rigid 

 body, one point of which is fixed at 0, the axes moving as in §§ 3, 5. 

 Since the absolute momentum of m in the position (^, ri, Z) at time t is 



w. {: (">2 + ^2) —V {'''3 + ^3)} along 0'^, etc., 



