VOL. XXXVir.J PHILOSOPHICAL TRANSACTIONS. 521 



This equation determines the sections of every spheroid, whatever be the 

 power of the distance by which the attraction acts ; excepting only the hypo- 

 thesis in which the attraction is in the simple ratio of the distance from the 

 centre inversely. 



In that case the general quantity 

 (^^-z = — '■ — '-] Gg will be ( ~ ~ — '■ — ) Gg, whose fluent can only be 



\ CA" CA y » V CA CA / *' •' 



had by logarithms, and gives p. ca log. CG — '^-^^^ = A ; or for the whole 

 weight of the column palog. r — '^-^ = a. 



To correct this equation, when h = 1, then must r =. a, and then pa\og. 

 a — -^fa = A; hence the equation corrected is jba log. r — ■^— ^ =r palog. a 



— —, or 2 pa log. - ='LJ1 fa;or taking the natural numbers of these 



logarithms, and putting c =: the number whose logarithm is 1, then is had 



r z= ac ^ v-'-' ^r. 



Hence it appears that the meridians of the spheroids will always be algebraical 

 curves, except only in this last hypothesis. 



If the equation of all these curves be desired in the usual manner, by per- 

 pendicular co-ordinates, it may be easily done, thus : taking ce = x, and 

 DE =: y, there will be r* = x'^ -\- y'-, and hr = t/. Then, exterminating h 

 and r from the general equation, there will result 



ipix' + /)'-" + ^ - {n + 1 )/«"-'/= (2/) - nf-f)a'' + '. 



(£iL - L) 

 And when tz = — 1. then x^ -\- y- ■= arc^P"" f' . 



But our former method, of defining the curves by radii and angles, is as 

 well, and perhaps more commodious than that which defines them by co- 

 ordinates. 



Though h be treated as a variable quantity, yet it varies not beyond certain 

 limits, which are O and 1 ; so that our radial equation will only define the part 

 of this curve whose amplitude is a right angle ; but when those curves consist 

 of four similar and equal arcs, the whole curves of all the meridians will be 

 given by our equations. 



It will now be easy to determine the ratio between the two arcs of the section 

 in any hypothesis. 



Since the general equation is Ipr" + ' — (?<+ l)/A^;-^a" - ' = {1p — rif— J) 

 a"+ ' \ to find r when A = O, it will be ipr" + ' = {2p ~ nf — f)a" + ' ; fi-om 



I I 



whence results ca : cp :: (2/j) " + ' :(2p — nf — /)" + ' . 



VOL. VII. 3 X 



