of a rigid Body round affixed Point. 1 7 



I rf 1 TT , . rd? , ^ / 



— ■ —j— . ^2* Hence &>' cos 8 = •— / -p^- ; but as &> = -=p- 



dcd _ „ c?P 



-^ - -y-pr- 



We shall have cJ cos 8 = —r-r (36.) 



dt 



There are many curious properties of rotatory motion, a few 

 of which only are subjoined, our paper having already much 

 exceeded its limits. 



XXVIII. Let segments equal to I, measured from the cen- 

 tre, be assumed on the three principal axes of rotation, the sum 

 of the areas described by the projections of these lines on the plane 

 of the impressed moinent, varies as the time. 



Let Sc be the area described by the projection of a portion 

 of the axis of c, equal to /, on the plane of the impresseil 

 moment ; then the projection of I on this plane is I sin y, and 

 the differential of the area 



-^ ^l sm y-j^-, (37.) 



d 1^ 

 or substituting for sin y and ~-' their values given by (8.) 



and (26.), we find 



In like manner, 



dt V « J dt \ b- J 



Adding these equations together and integrating, 



Sa + Si + Sc = 2l-vjt + constant. . . . (39.) 



XXIX. Should the portions I, instead of being equal, be 

 proportional to the square roots of the moments of inertia 

 round the corresponding axes, the sum of tlie areas described 

 by the projections of these lines on the plane of the impressed 

 moment still varies as the time. 



C • M^ 

 In (38.) let /"^ = — tj = — :t c^, where m is a constant rijjht 



line, and equation (38.) is changed into 



tf = ^(^'-~-^) (-•) 



Phil. Map, S. 3. Vol. 20. No. 128. Jan. 184.2. C 



