in the interior Parts of the Earth. 325 



torial radius, q the density, and e the ellipticity, of any in- 

 terior stratum ; then, tt being the circumference when the 

 diameter is unit, the gravitation at the surface of the stratum 

 will be proportional to 



a- 



Let g denote the space through which a heavy body falls in 

 a second at the equator, and ?h the mean density of the earth, 

 that is, the density of a homogeneous sphere equal in bulk to 

 the earth and attracting with the same force ; then, 



^-Ttf^ d- da A "^ in- 



g ^ _ _ ___^ 



the integral being taken from a = to a = 1. Hence the 

 gravitation at the distance a from the centre will be 



Let w denote tbe centrifugal force at the equator, that is, 

 the versed of the arc of the equator that passes over the meri- 

 dian in a second by the diurnal motion ; then a co will be the 

 like force at the distance a from the centre ; and the propor- 

 tion of this force to the gravitation at the same distance will 

 be equal to, " v- ^"^"^ 



g 3fza"-da 



Now, put <$! = — = -Tgr- ; and assume, 



e = X X <^ X - . ., ■ , (1) 



and X will be the proportion of the ellipticity to the centrifu- 

 gal force, that force being estimated in parts of the gravita- 

 tion. At the surface we obtain {e) = [x) x <f, the symbols 

 {e) and [x) denoting what e and x become when a = 1 ; and, 

 at the centre, we have e"^ = x° x <p x m, the central density 

 being unit, and e° and x'^ denoting what e and x become when 

 a = 0. 



Now, if we observe that the symbol A in Clairaut's equa- 

 tion is equivalent to -.-, that equation may be thus written, 



Sa^efqa^da =/gd{a'e) + -^ a' + a'{F -f^de), 

 all the integrals beginning when a = 0, and F being the whole 



* No distinction is made here between the gravitation and the attrac- 

 tive force at the equator, nor between the gravitation to a spheroid of 

 small oblateness and a sphere of equal equatorial diameter ; because the 

 quantities are to be used in an equation between the cllipticities and the 

 centrifugal force, and the method of Clairaut allows the squares of these 

 small quantities to be rejected. 



integral 



