GRAVITATIONAL STABILITY OF THE EARTH. 181 



This equation is a linear differential equation of the second order ; and the forms of 

 the coefficients show that the point r = is a critical point, and that there is no 

 other critical point at any finite value of r. If we seek a solution in series of 



the form 



/. = 



we find the " indicial equation " 



m(m-l)+2(n+l)m = 0, 



from which either in, = 0, or m = (2n+l). We must take the series for which 

 m = 0, because ?"/ must be finite at r = 0. Further, the form of the equation shows 

 that this series contains even powers of r only. We assume, therefore, for f n 



the form 



/ = A [H-a,r*+a 4 r 4 +... + a fc r* 1 +...], 



where A is an arbitrary constant, and then we find the sequence equation 



= 0, 



or 



h' 



fc+ ' = a "(2K+2) (2K+2n+~3) 

 Hence we have 



f - . A fl - *' t^J-AKr*. itfi!L+6) Q_{ A'_(n 8) *} 4 _ 

 / ' L 2.(2n+3) 2.4(2n+3)(2n + 5) 



. y {A'+(n+6)<'}{y+(n + 8)a'}...{A'+(n + 2* + 4) J i a } ,. 1 

 2.4...2c(2n-l-3)(2n-f5)...(2n + 2K+l) "J" 



The series is convergent and represents the function f n for all finite values of r. 

 9. We must next determine the function E. (r) from the equation 



or 



, ( . 



" 



r dr 

 or 



We introduce an intermediate function n (r) by the equation 



Then 



6. = \rf n dr 



. ......... (35) 



= C + A 4r--" ^r* + 



2.4.(2n+3) 2.4.6(2n-f3)(2n--5) 



(n-HS)^}^^ 

 )(2n--5) "J ' 



