44 Motions of a System of Bodies. 



they are also independent of any force which acts towards the ori- 

 gin of the coordinates, as was remarked of the motion of a particle 

 oC matter at p. 133, vol. xxii. If the forces cause finite changes in 

 the motions of the bodies in the indefinitely small time dt, (7) will 

 be changed to D.Smc=dt.SmTr, D.Swi,c = dt.Sm,T,r 9 D.Sm y/ c 

 dt.Sm„T„r t (8) ; which are also independent of any changes which 

 the bodies receive in their motions from their finite actions in the in- 

 stant dt, they are also independent of any finite changes caused by 

 forces, which are directed towards the origin of the coordinates. 



(7) and (8) are easily adapted to the motion of a solid by putting 

 M=to its mass, and changing m into dM = any indefinite element of 

 the solid, and representing indefinitely by c, the values of c, c', he. in 

 the first of those equations ; and by ,c, n c the corresponding quanti- 

 ties in the second and third ; and using S as the sign of integration. 

 If the solid revolves around the axis of z as a fixed axis, the second and 

 third equations in the case of (7) will evidently not exist ; also tfo= dv' 

 &c.=the angle described by the solid around the fixed axis in the 

 time dt ; and r, r', &c. will each be invariable ; .'.since c — r 2 dv 9 



d*v SdMTr 



c'=r' 3 dv, &c. it is easy to find, (by the first of (7),) ~^~^jWT J 



put SdMr 2 =Mk* =the moment of inertia of the solid around the 



d 2 v SdMTV j f . 



axis oi z, and there results ^ = "]u7T" > ( 9 ) J which formula is 



well known : in the same way by (8) when the forces are impulsive, 



dv 

 or cause finite changes in the time dt ; by putting D.-j-~dt=w 



the angular velocity of the solid, (caused by the forces,) we have 



SdMTr . . . ' 



w ~ Mk 2 ' ( 10 )" Again, smce a?=r cos. t>, y=r sin. t?, &c, by 



* V ^ n 1) Tn.li — 11 u 30 



putting P= -Tsin. v, Q=Tcos.t;, &,c. then c— ~^--= , , 



&c, also Tr = xQ — Vy, he.; .'.the first of (7) becomes 



xd 2 y —yd 2 x 

 Srol j— J ~Sm(xQ — Py), which agrees with the equation 



found at p. 66, Vol. i, of the Mecanique Celeste, and by making 

 similar changes in the second and third of (7), the two other equations 

 given at the place cited, are easily found : I would also observe that 

 the same equations may easily be found by (1), (2) and (3), but I 

 have preferred the method which I have used because it has some 

 advantages over the other. 



