428 APPENDIX. 



odic 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 



point being r, the force 3 ; to the other point the radius 



m 



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

 f 



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 * m t At every other 



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

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



taken = - -% , and Pa = - 2 , the resultant at that point 



P bisects act! and is P c, and produced, P M cutting the 

 axis. From hence may be seen how complicated would be 

 the analysis, how next to impossible the geometrical con 

 struction of the locus of P, by referring the lines PM to 



