VELOCITIES, &c. WITH RESPECT TO AXES MOVEABLE IN SPACE. 3 



It is well to observe that ^ vanishes, when the three axes 01, OU, OP lie in the same 



at 



plane, and in particular when two of them coincide, as is evident from the above equations, or 



from considerations similar to those by which it was obtained. 



4. In the above investigation we have supposed u to be constant both in direction and 

 intensity ; let us now suppose u to vary in both respects with the time {t). The change in u 

 in the time dt may be conceived to arise from its composition with the quantity fdt in the line 

 OF, and / may properly be called the acceleration of u at the time t. Now fdt may be 

 resolved in the plane UOF into /. dt cos FU along OU and /. dt sin FU perpendicular to OU, 

 and the components oi u + du will therefore be u + f. dt cos FU along OU and /. dt sin FU 

 perpendicular to OU; whence, if d(p denote the angle through which OU shifts in the time dt 

 towards OF, it will readily be seen that ultimately 



^=:f cos FU, u^=fsmFU...{B), 

 at at 



If then the acceleration / be known both as to direction and intensity at every instant, the 

 motion oi OU and the variation in the intensity of u may be determined by these last 

 equations. In fact, the point U on the sphere of reference continually follows the point F 



with the velocity —-, so that the problem of determining U's path is the same as the old 



problem of the path described by a dog always running towards his master who is himself in 

 motion, the only difference being that the path is here on a sphere instead of a plane. 



5. Next for the variation of Up, when u varies with the time. It is plain that Up varies 

 from two causes ; first, by reason of the acceleration /, and secondly, by reason of the motion 

 of OP due to the angular velocity Q about 01, and that the total variation will be the sum of 

 these two partial variations. Now the latter has been calculated above, and the former is 

 obviously the resolved part oi fdt along OP or f.dt cos FP, therefore we obtain the equation* 



^^^ = f cos FP - uQ sin lU. sia UP, sin lUP.. .(C). 



This equation of course contains the previous equations (B) : thus, if OP and 0Z7 coincide 

 always, UP is always zero and the second side of (C) reduces to its first term : and again if 



OP be always in the plane FOU and perpendicular to OU, Up is always zero, Q = — - , lU 



and UP are quadrants, lUP a right angle, and FP the complement of FU, and therefore, as 

 above, 



= fsinFU-u-^. 



'' dt 



6. We may farther illustrate the application of equation (C) by supposing OP to coincide 

 with certain other lines specially connected with 0?7 and OF. , , 



" It should be remarked here that the angle lUP must be I Q about 01 causes the motion of P, resolved in the arc UP, to 

 considered positive or negative, according as the positive rotation I be from or towards U. 



1_2 



