Bessel- Clifford Function, and its applications. 513- 



Figure of the Earth. 



11. The investigation may be resumed here in the 

 ' Quarterly Journal of Mathematics,' vol. xvii, 1880, of 

 the " Differential equation of the ellipticity of the strata 

 in the theory of the Figure of the Earth/' where a 

 generalisation was made of Laplace's assumption, leading 

 to differential equations of the form we have been con- 

 sidering. 



The differential equation for € the ellipticity of a stratum 

 of a liquid gravitating sphere, stratified originally in con- 

 centric spherical surfaces, and disturbed slightly by a solid 

 spherical harmonic of order i, is given there: 



r^ dMe_ tori* dp 



Me dr 2 -^ + 1 J+ M di 



•'(•■ + 1)+™- -£■ • • • CD 



where M denotes the mass of liquid inside a spherical 

 stratum of radius r, so that 



-fc =*™*P (2> 



Laplace solved the equation for the Figure of the Earth,, 

 where the disturbance from the spherical form is due to the 

 rotation about the polar axis and ?' = 2, on the assumption of 



4tt?- 2 dp , , 1 ,_ 



"IT £ a collstallt = ~~« 2 ' su PP ose > • • • (•*) 



the negative sign being introduced, as -y- requires to be 

 negative for the stability of the strata. Then 



( 



dr? dr " ' dr 2 ' dr 



^ + 2^+^ = 0, ^f^ + SP-O, . (5) 



dr 2 dr a~ r dr- a~ 



' P = Asin (a + 6 ) (,;i 



in which e = 0, to make the density finite at the centre. 

 Then, in Laplace's notation, writing q for , 



sin qr dp 

 qr 

 M = — 47r/o a 8 (sin qr — qr cos qr) (8) 



dp l cos qr sin nr\ 



