248 Variations of the Arbitrary Constants in Elliptic Motion. 



the Vulcanian, as to render each capable of assisting towards the 

 development of the other, for that a great portion of the present 

 state of the crust of the earth has resulted from the agency of heat, 

 no one in possession of the facts, and exercising a sound judgment, 

 will now attempt to deny, and also that the agent which caused 

 these appearances is still in active operation. The trident of Nep- 

 tune would never have been reared over his oceans, had not Vulcan 

 forged it for the Sea King, yet it is ever to be borne in mind that the 

 liquid realms of that monarch contain in their constitution the prin- 

 ciples over which in another mode of combination, the master of fire 

 rules with equal power. 



(To be continued.) 



Art. IV. — Theory of the Variations of the Arbitrary Constants 

 in Elliptic Motion ; by Prof. Theodore Strong. 



We shall (for simplicity,) consider a system of three bodies only, 

 whose dimensions are so small when compared with their distances, 

 that they may be neglected, so that their masses may be regarded 

 as united at their centres of gravity. 



Let then M', m, m' denote the masses of the bodies collected at 

 their respective centres of gravity, and suppose that m, m! revolve 

 in the same direction around M' considered as at rest, and that the 

 bodies attract each other with forces, which are expressed by their 

 quantities of matter divided by the squares of their distances ; to de- 

 termine the circumstances of the motion of m. 



Put t for the time, and x, y, z; x' , y', z' for the rectangular coor- 

 dinates of m and w', when referred to any fixed system of rectangular 

 axes whose origin is at M', also r, r' , r" severally for the distances 

 of 7n, m! from M, and for their distances from each other, at the time 

 t\ then r2=a;«+y2+z% r'"-=x'^--\-y'^ ^z'\ r"^ ={x^—xY-^ 



{y'-yYH^'-^Y^ (i). 



M.' m m' m' 

 By supposition, — j — > -7^? ^777 are the forces with which the 



bodies attract each other, and if di denotes the differential of the 



M' m m' m' 

 time, it is evident that —^dt, ~^dt, -j^dt, -^dt will denote the ve- 

 locities which the attractions considered as constant will communi- 



