GRAVITATIONAL STABILITY OF THE EARTH. 

 Hence the equation 



) = 

 becomes 



(99) 



27 L ^ ^ J 



where x is written for sa. 



33. An irrelevant factor v has been introduced into the left-hand member of (99). 

 We find that when x = this expression becomes 



which is positive for all admissible values of v. We find also, by the method of 

 asymptotic expansion, that the expression is positive, when x is great, for all values 

 of i/. We proceed to show that, in the important cases v = and v = , the equation 

 has no real root. The left-hand member of (99) being an even function of JT, we may 

 treat x as positive. 



When v = 0, the left-hand member of equation (99) is 



which is positive for small values of x. The differential coefficient of the expression 

 within the square brackets is 



1 4* xV 1 ** dx + 5zV*, 



o 



which is positive for all real values of x. Hence the left-hand member of 

 equation (99) cannot be negative if x is positive, or the equation has no real root. 

 When v = J, the left-hand member of equation (99) becomes 



**]*5^-* (IOO) 



The expression within the square brackets is greater than 



(101) 



where this expression is obtained from the other by replacing every positive coefficient 

 by the next smaller integer and every negative coefficient by the next greater 



integer. 



VOL. CCVH. A. 2 K 



