313-314] MACLAURIN'S ELLIPSOID. 583 



Since a*/( a2 + ^) is greater or less than c*/(c* + X), according as 

 a is greater or less than c, it follows from the forms of , 7,, given 

 in Art. 313 (5) that the above condition can be fulfilled by a 

 suitable value of n for any assigned planetary ellipsoid, but not 

 for the ovary form. This important result is due to Maclaurin*. 



If we substitute from Art. 313 (11), the condition (6) takes the 

 form 



= (8g + l) foot-' r-3?" ............... (7). 



The quantity f is connected with the excentricity e of the 

 meridian section by the relations 



The equation (7) was discussed, under slightly different forms, 

 by Simpson, d'Alembert-(-, and (more fully) by Laplace*. As 

 f decreases from oo to 0, and e therefore increases from to 1, the 

 right-hand side increases continually from zero to a certain maxi- 

 mum (-224)7), corresponding to e = '9299, a/c = 2-7198, and then 

 decreases asymptotically to zero. Hence for any assigned value of 

 n, such that n 2 /27r/> < -2247, there are two ellipsoids of revolution 

 satisfying the conditions of relative equilibrium, the excentricity 

 being in one case less and in the other greater than '9299. 

 If n*/2irp > '2247, no ellipsoidal form is possible. 



When is great, the right-hand side of (7) reduces to T 4 5 - 2 approximately. 

 Hence in the case of a planetary ellipsoid differing infinitely little from a 

 sphere we have, for the 



If g denote the value of gravity at the surface of a sphere of radius a, of the 

 same density, we have g = ^ IT pa, whence 



Putting n*a/g = %&$, we find that a homogeneous liquid globe of the same 

 size and mass as the earth, rotating in the same period, would have an 

 ellipticity of 5 J T . 



* I. c. ante pp. 322, 367. 



t See Todhunter, Hist, of the Theories of Attraction, etc., cc. x., xvi. 



$ Mecanique Celeste, Livre 3 me , chap. iii. 



