Motion of a System of Bodies. 53 



dx' 



% (*') » (since '^ > &c. are each=0.) if we suppose a/=0, y'=0, 



-v/rfo? a +dy* +dz^ 



we shall have zVp 2 +? 3 ~ jf* =the velocity of a 





point which is on the axis &IV, at the distance *' from the origin, also by 

 (47), — ;- */ \ __ c 2 =: the sine of the ande made by the axis 



oi z with the momentary axjs,. v , — -^ =~= = the perpendicular 



from the extremity of 2/ to the instantaneous axis; put w=the angu- 

 lar velocity around the instantaneous axis, and we have z'y/p* + q 2 

 z'x/p*+q* 



By (47) and (48), p=a,w, q—b,w 9 r = c / w, (49), where p, g, r 

 are evidently the momentary rotations around the axes of #', y', z' 

 respectively ; hence it is evident that rotary velocities are compound- 

 ed and resolved by the same rules as rectilineal velocities. 



Remarks. — It is evident that if the origin of the coordinates is at 

 the centre of gravity of the system, all the formulae which we have 

 found will apply, whether the centre is at rest or in motion ; for (19) 

 which have the same forms as (18), are applicable whether the cen- 

 tre is at rest or in motion j hence by proceeding with (19), as we 

 have done with (18), we shall obtain the same results as before ; .'.by 

 placing the origin at the center of gravity, all the above formulae will 

 apply when the system is free ; and the motion of the centre will be 



found by (4). 



Again, it is supposed in (kf) that the bodies are so small, that x\ 

 y', zf may be considered as having the same values for all the points 

 of each, but should not this be the case, we must change m into dm, 

 then find the value of A for each body, by taking the integral relative 

 to its mass ; then the sum of all the values thus found, will be the 

 complete value of A ; and in the same way we must find the com- 

 plete values of B, he. ; but should the system be a continuous sol- 

 id, we must find the values of A, B, kc. by integrating relative to 

 its mass. 



