520 PHILOSOPHICAL TRANSACTIONS. [aNNO 1732. 



spheroid through the axis. Now since the parts of the fluid are at rest among 

 themselves, every column cd will have the same weight towards c ; consider- 

 ing therefore one column cd, which makes with cp a given angle, whose sine 

 is = h, to the radius 1 , and which is composed of an infinite number of small 

 cylinders ; I find the weight of any small cylinder towards c. 



Since the absolute gravity at a is given and = p, to have the gravity at g, it 

 will be p : p^ :: ca" : cg" ; hence is the gravity at g, or /j' = — ^. 



But since, because of the rotary motion, any part of the fluid is repelled by 

 the centrifugal force in the direction gh ; and since, in motions arising from 

 contemporaneous rotations, the centrifugal forces are as the radii of the circles 

 described; if the centrifugal force at a be given and =^ f, to have the centri- 

 fugal force at g, it will be/:/'' :: ca : lg = (because lg : cg :: A : 1) /;. cg ; 

 hence the centrifugal force in g, or/' = - ' : but this force, since it acts in 

 direction gh, is decompounded into the two forces gk, kh, of which gk is the 

 only part acting in the direction cg. Therefore this force gk will be had by 

 saying as gh : gk. or as 1 : A :: — ^ — '■ : /' = • — ~ = the force of the small 

 cylinder g^ towards d. Therefore the force of that cylinder g^ towards c, 



will be only ^^^ — —; and the weight of that cylinder towards c, 



will be Pl:^ — -i-LjIflj g^. Now since g^ is the element of cg, therefore 



the integral will give ^-^ — ' '^, "Z^" for the weight of the column cg ; 



and ^-^ '-^ — for the weight of the whole column cd, which ousht to 



M+l.CA" CA ° != 



make a constant weight a. 



If therefore there be called ca = cr, cd = r, it will be — ■-^-^ = a. 



n + \ .a" 2a 



And since this equation, whatever be the value of A, will always obtain, if A be 

 assumed as indeterminate, the preceding equation will give the relation between 

 any radius CD and the sine of the angle it makes with the axis pq. 



Now the constant quantity a is to be determined. That the preceding equa- 

 tion be adapted to the section of that spheroid whose semiaxis is ca = a, when 

 DCP is a right angle, or when A = 1, then will r = a; hence it will be 



pa" + ' fan 2p — w/'— /' 



-i- ■!-— = A, or A = -i- ■- — ^-a. 



II + l.a" la ' 2.n + 1 



And thus the enuation corrected will be — — ; — - — — - — = —, — = -a, 



' )i + I .a" ^a '2 .n + I 



or 2pr+ ' - (n + \) fliY-W" = {ip - nf -/) «" + '. 



