<i42 CELESTIAL MECHANICS. 1. VU. 25. 



S (x'ddy+xddy — y'ddx — yddx') Dm—ddy Sx'Dwt — ddjc 

 Sy^i)9?i+x S ddy'Dm—y Sddo.' D/w ; each of the terms of 

 the second member of this equation being equal to no- 

 thing, by the properties of the centre of gravity, the 



equation "(1)" will become S "^fY Tim=:S {Qx 



— PyXBm), since the parts [2] and [3] destroy each other 

 in consequence of the equation (J) of the preceding pro- 

 position ; and the fluent of this expression, considered with 



respect to the time ^, gives us S — a. — Bm^iSj^ (Qx' 



—'Py')dtBm. 



Scholium 1. These three equations include the prin- 

 ciple of the constancy of the areas described ; they are suf- 

 ficient to determine the rotatory motion of the body, round 

 its centre of gravity, and in combination with the three 

 equations of the preceding proposition, they afford us the 

 complete determination of the progressive and rotatory 

 motion of the body. 



Scholium 2. If the body is attached to a fixed point 

 with liberty to move round it, the motions may be deter- 

 mined by means of this proposition, as is obvious from 

 article 308 ; but in that case the coordinates x' , y\ and z' 

 must be supposed to originate at that point. 



344. Definition. The three principal 



axes of rotation of any body are those, with 



respect to which the three sums ^xy'iym^ So!' 



zjym^ and Sy'zTim vanish, x\ y\ and z" being 



axes moveable with the body. 



[Scholium 1. If the body revolve about a:'', the sum 

 Syyom will be the effect of the centrifugal force of all 



