NEWTON S PRINCIPIA. 145 



- , S O being put = c, and AD = x t and 



\ x a ) 

 A S = a, as before ; fy d x may be found as before by 



(x of d x c 2 d x 



integrating - -f - - . The fluent is 

 \ x a ) 



(* ~ a) 3 &quot;&quot; , (x-aj~ n p , p 2 c 3 -&quot; 



-- c - - 1 -- {_ L, ; an( i (^ = _^ - - . 



3 n 1 n n- 4 71 + 3 



and the whole attraction of the segment upon the particle 



at the centre S is equal to - -- - + -5 ; 



3n l ?i n 2 4:71 



/ \ n 



Thus, if n 2 the attraction is as , or as OB 2 



directly, and as S B inversely ; and if c = o, or the at 

 traction is taken at the centre, it is equal to a ; and if 

 the attraction is as the distance, or n = 1, then the 



attractive force of the segment is - (a 2 c 2 ) 2 . 



ii. Our author proceeds now to the attractions of 

 bodies not spherical ; an inquiry not perhaps, in its 

 greater generality, of so much interest in the science of 

 Physical Astronomy, where the masses which form the 

 subjects of consideration are either spherical, or very 

 nearly spherical, to which our examination has hitherto 

 been confined. But this concluding part, nevertheless, 

 contains some highly important truths available in astro 

 nomical science, because it leads, among other things, to 

 determining the attraction of spheroids, the true figures 

 of the planets. 



The attractions of two similar bodies upon two similar 

 particles similarly situated with respect to them, if those 



attractions are as the same power of the distances -, are 



to one another as the masses directly, and the ?* th power 

 of the distances inversely, or the ?* th power of the homo- 



L 



