1 90 Prof. Sylvestei' on the Motion and Best of rigid Bodies. 

 Also if X^ Y, Z be the impressed forces, we have 



x, + x = 4f ,.) 



Y, + Y = ^ (r,) 



Z,+ Z=^- (6,. 



And by Gauss's principle, calling m the mass of the particle 



ator^j/, 2r, A5'?«. {X;^ + Y^^ — Zj] = 0. 



Hence equating separately to zero the coefficients of a^ hj c^ 



and of «^ ^j 7; in the quantity Xm (X^ A X, + Y^ A^ + Z^A Z^) 



we have 



5m.X, = n 



JS" ?« . Y^ = 



JS-w.Z, = ' ._ 



^7« {Z^y-Y,z) =0 ^ ^ 1^ 



^?«.(Y,a:— X^?/) = I 



5'»2.(X^S-Z;.r)=: J 



Lastly, we have the = °^ 



u=^ (13) 



d7/ 



~dt 



dz 



dt 



(14) 



(15) 



From the fifteen equations marked 1 to 15, the motion 

 may be determined by assigning the position of each particle 

 at the end of the time t in terms ot its three initial coordi- 

 nates, its three initial velocities, and the initial values of the 

 nine quantities. 



i' 711 .X Xmyz Xm x- 



Xm .y Xm z X X mif 



Xvi.z Xm X y Xm z"^ 



In the case of rest X^ = —X Y^ = — Y Z; = — Z, and 



the = "s 7 to 12 inclusively taken, express the conditions of 

 equilibi'ium. 



The equations o, jh ^■> which have been obtained from con- 

 ditions ^jwe/j/^f'o???^/^?*:^/, establish the well-known but inter- 

 esting and not obvious fact, that any small motion of a rigid 

 body may be conceived as made up of a motion of translation 

 and a motion about one axis. 

 University College, London, Dec. 8, 1838. J. J. SYLVESTER. 



