316 VII. DYNAMICS OF ROTATING BODIES. 



Since the last terms of both these equations contains &', it is suggested 

 that we change the independent variable from t to #, which is done 

 by dividing through by #', giving 



226a) (Ma 2 -f C) ~ (y f -f cos # - ^) - Ma 2 sin & tf = 0, 

 b) ^{(Ma 2 + C) cos & <y + cos# #') + ^.sin 2 ^.^'} 

 -f ^fa 2 sin^- cp f = 0, 



as two simultaneous equations to determine ^' and qp' as functions 

 of -9-. Now observing that <p ' -f- cos # ^ f = r, let us multiply the 

 first equation by coso)* and subtract it from the second, obtaining, 

 on performing the differentiations and simplifying, 



227) - Csin# - r + J.(sin 2 # ^') = 0. 



Introducing the value of ^' from 226 a), performing the differentia- 

 tions, we have finally, 



d*r . . dr MCa* 



228 ) 



which is the same as the equation 210) obtained by Appell and 

 Korteweg, by a totally different process. 



Having obtained r, we obtain iff' from 227), we may then obtain 

 #' from the equation of energy, and obtain the time as before. 



1O2. Moving Axes. It is often convenient to refer the motion 

 of a body to a set of axes which are themselves moving in space. 

 Let us first suppose that they move parallel to themselves and that 

 the moving origin has the coordinates fj, ??, with respect to a system 

 of parallel axes fixed in space. Let the coordinates of a point with 

 respect to the fixed axes be x\ y\ s* and to the moving axes x, y, z, 

 then 



229) 



dx' di- dx d*x' d 2 | d*x 



~dt = ~~dt~T~ lit' ~di* ~ dt* + dt* 

 dy' dr\ dy d*y' d*r\ d*y 



' 



dt ~ dt ^ dt dt* ~ dt* r dt* 

 dz' d dz d* z' d* d*z 

 lit == lli + ~di' ~dt* == dt* "*" dt*' 



showing that the velocity and acceleration of a point with respect 

 to the fixed axes are the resultants of the velocity and acceleration 

 of the point with respect to the moving axes, and of those of a 

 point rigidly connected to the moving axes. We may accordingly 

 consider the moving axes at rest, provided that, in addition to the 

 forces impressed upon the system, we impress forces capable of 



