394 ' Mr. George J. Allman's Account of 



^ = ;:73+2-^7T-(l-3^in^n (13) 



C — A 



P = 3 — 7^- cos \ sin X. (14) 



The direction of the force Pis towards the plane of the equator ; this appears 

 from the shape of the " ellipsoid of gyration," which in this case is a prolate 

 surface of revolution. 



Piiop. IX. Clairaut's Theorem. 



Whatever he the law of variation of the earth's density at different distances 

 from the centre, if the ellipticity of the surface be added to the ratio ivhich the 

 excess of the polar above the equatorial gravity bears to the equatorial gravity, their 



5 



sum will be „ q, ivhere q is the ratio of the centrifugal force at equator to the equa- 



torial gravity. 



For suppose the attracted point M to be on the surface of the earth, which 

 is known to be an oblate spheroid of small ellipticity. Then, from the pnn- 

 ciples of Hydrostatics, since the tangential force is cypher, we have 



BcosO - PsiuO - w^r cosX cos (6 - \) = 0, (15) 



where w denotes the angular velocity, and 6 the angle which the tangent to 

 the meridian through the attracted point makes with the radius vector ; de- 

 veloping cos {0-\) and arranging, we obtain 



(R - a.Vcos-\) cos0 = (P+ u?r cos\ sinX) sin a (16) 



But from the property of the elliptic section made by the plane of the meridian, 

 we have 



i sin \ cos X „ . ^ 



cot B — -^ T jr- = 2e sm X cos X, q. p., 



1 - r cos-X ^ 



wliere e is the excentricity and e the ellipticity of this ellipse. 



Substituting in (16) this valueof cot 0, and the values oi R and P from (13) 

 and (14), the equation of equilibrium becomes 



