OF THE MOTIONS OF A SOLID BODY. 



277 



z'^) T>m, m the mass, and h the distance of 

 the centre of gravity from the axis of motion. 



The preceding investigations are sufficient for determin- 

 ing the motion of a solid round its centre of gravity, when 

 it is either at liberty, or fixed to a single point of suspen- 

 sion only : it now remains for us to consider the motion of 

 a solid round a fixed axis. 



We may call the axis of motion x\ and suppose its di- 

 rection to be horizontal : the last of the equations {B) 

 (343), will be sufficient to determine the motion ; that is 

 gYdzWd/^ Dm=S/(jRy— ^2') dtDm=N'\ We may 



suppose y' to be also horizontal, and z' vertical, or per^ 

 pendicular to the horizon, the plane of y' and z' passing 

 through the centre of gravity of the body, and a moveable 

 axis being supposed to pass constantly through this centre 

 and the origin of the coordinates. Now Q being the angle 

 which this new axis makes with z\ and y" and if' being 

 the coordinates perpendicular and parallel to this new axis 

 in the plane of y and s^, we 

 have y '=■]!' cos fi + z'' sin 6, 

 and zl—z!' cos d — y" sin 5, 

 consequently [y'dz' — zf^y'zz 



d5 \ {y" cos fi + 2'' sin &), (— ^ 



z" sin &—y" cos &)—{z" cos ^ 

 —y sin &). {—f sinfi-fz^' 



cos Q)\ =:d5 I — (/ cos 6 



+ /' sin d) 2— (2'^ cos d— / sin 6)^ | = -dd f f^cosH-^-sinH) 



+ z"^ (sin H + cos H) \ and] S i^l:Zl^ Dmzz-^ S 

 3 d^ oj 



