107 



equal to ^p - ^- n9 and, in like manner, it will attract 



M M n+1 



E with a force equal to jp if it be equal to.-^ ^. n . 



If n = 2, or the force be as the inverse square of the 

 distance, the body placed in the centre will be equal to 



M 3 



Tp - MV2&amp;gt; if n ~ ~~ 1* or the attraction be directly as the 



distance, the body will be in both the case of M and 

 E equal to E + M ; and if the attraction be as the square 

 of the distance directly, the central body will be in the 



-&quot;. , .. (M + E) 2 , (M + E) 2 



two cases of the two bodies, * ^-p and v - ^ - 



IVI jit 



respectively. 



Xext as to the absolute trajectory of the bodies thus 

 acting on one another, or their path in space, we have 

 an investigation analogous to those inquiries formerly in 

 stituted where the centre of forces was fixed. For the 

 body or bodies being known (by what we have last shown) 

 whose mass gives at the centre the same attractions as the 

 two bodies exercise on each other, we can determine for 

 each of these bodies the path in which it will move, pro 

 vided we know the initial direction and velocity. Thus 

 let m = 2 in the last expressions, we have for the mass by 

 which Mis attracted towards the common centre of gravity 



E 3 



j^f - pY 2 ; and proceeding as was formerly shown in the 



case of immoveable centres, we find that if the curve de 

 scribed round the centre at rest be a circle, if that centre 

 moves in a straight line, the orbit in space will be of the 

 cycloidal kind; if the centre moves in a circle, it will be 

 an epicycloid or hypercycloid ; and if the curve be a conic 

 parabola, the motion of the centre will change this into a 

 cubic parabola, which will thus be the path arising from 



