194 ON FORCES OP ATTRACTION 



I is the initial velocity, or velocity of projection, it will 

 describe the same ellipse, provided I 2 = V 2 -f- u 8 . 



5. The other theorem follows from his integration which 

 gives the expression for the time. It is that if two equal 

 forces act upon the body directed to the two foci, and the 

 masses of the attracting bodies consequently are equal, the 

 revolving body will describe the ellipse in a shorter periodic 

 time, will move more swiftly, than if the whole mass were 

 placed in one focus and acted from thence upon the revolving 

 body. He takes the example of the tangent of the orbit with 

 the axis making an angle of 30, and finds the periodic time 

 shorter in the proportion of nearly 78 to 100 when the 

 attracting mass is divided into two, one acting in each focus, 

 than when both combined act from one focus. 



6. What renders this problem of more centres than one so 

 difficult, is that the resultants of the forces pass through 

 different points, and that they vary by a law which differs 

 in each case, as the locus of their extremities is a different 

 curve. Take the least complicated, but still full of diffi- 

 culty, that of two fixed points as the centres of force, and 

 take the instances in nature of the forces being inversely 

 as the square of the distance ; the radius vector to one 



rrn 



point being r, the force ; to the other point the radius 



171' 



vector q. the force ^. Now the force which acts on the 



<? 



body being the resultant of these two, and these forces not 

 being as r and q, the diagonal does not pass through the 

 middle point of the line joining the two centres ; except in 

 the single point of the orbit where r = q, and even then it 



may not reach that line, for it is ^ . At every other 



point it runs in a different direction. Let S and S' be the 

 two fixed points ; S P = r, and S' P =. q. Then P a being 



77X 7TL 



taken = . and Pa' = -5-, the resultant at that point 

 r 2 q* 



