424 APPENDIX. 



with the distance increasing, as in -^ and -^ combined 



, 



r q 



with - and -, than that one set should decrease and ano- 

 r q 



ther increase with the distance, as in and -~ with r and 



r 2 cf 



q, in which case there must even be an extinction of force at 

 one point, where (taking the sum of the forces instead of 



.! . T , m m r-\- a 



their resultant)- 2 H -- : 2 ~ - ~, or r is as in the equation 



Cl 77? 



f 3 + - 2 r2 = m *- Of course the value of q would be 



the same ; and the resultant (more accurately taken to 

 measure the increase of the force) would at one value give 

 the two sets of forces as counterbalanced. 



The younger Euler (J. A. Euler) has a paper in the 

 Berlin Mem. 1760, p. 250, upon the action of a central force 

 decreasing as the distance, in the case of the attracted 

 body s descent towards the centre, and states the reason of 

 this problem being insoluble except by arcs or logarithms. 

 He finds that taking a= the height from which the descent 

 begins, f = that at which the centripetal force is equal 

 to the gravity of the attracted body, the time of descent 



I /^* 



e is = - - / d y , . . ,. 



A/? / - ~ V being the dis- 

 J*J a 



^lo~ 



towards the centre is = 



A/? - ~ 

 J*J 



^ 



tance from the centre. 



No. IV. 



CENTRAL FORCES TO MORE THAN ONE POINT. 



1. IT is to be lamented that Sir I. Newton did not treat 

 the problem of forces directed to more fixed points than 



