313-814] MACLAURIN S ELLIPSOID. 583 



Since a?l(a? + \) is greater or less than c 2 /(c 2 4- 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 



-(3f*+l)?cot-&amp;gt;f-8f ............... (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 J. 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 (-2247), corresponding to e = 9299, a/c = 2*7198, and then 

 decreases asymptotically to zero. Hence for any assigned value of 

 n, such that n*/27rp &amp;lt; 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 9 /2irp &amp;gt; &quot;2247, no ellipsoidal form is possible. 



When is great, the right-hand side of (7) reduces to T 4 - 1~ 2 approximately. 

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

 sphere we have, for the ettipticity, 



&amp;lt; = (a-c)/a = K-2 = if^ .............................. (i). 



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

 same density, we have &amp;lt;/ = ^7rpa, whence 



Putting ?*%/#= say 5 we nn d that a homogeneous liquid globe of the same 

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

 ellipticity of TJ| T . 



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



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



Mecanique Celeste, Livre 3 mc , chap. iii. 



