GRAVITATIONAL STABILITY OF THE EARTH. 195 



and the equation may be written 



-5)-( I -|-))]^"f. >& = -- < 80 > 



22. The left-hand member of equation (80), being equal to 



is positive when x = 0. To determine its sign for large values of x we observe that 



and therefore there is an asymptotic expansion* for xe~ |x * I e**'dx when x is large in 

 the form 



. 5ar"+3 . 5 . 7x~ 8 + ... . 



Hence the expression in the left-hand member of (80) is asymptotically equal to 



The term of highest degree independent of v is 2x~* ; the term of highest degree 

 containing v is SKC'*. It follows that the expression is always negative when a; is 

 sufficiently great. The expression therefore changes sign for some positive value of x, 

 and the equation (80) has at least one positive root. 



23. When v = the equation (80) becomes 



If x* < 1 the left-hand member is necessarily positive. We shall take x* > 1 and 

 write the equation 



XT 1 



Let y denote the left-hand member of this equation. Then we have 



dx 



Since this expression cannot vanish, the equation cannot have more than one positive 

 root. 



* For the suggestion that this stop might prove useful in demonstrating the existence of a root of 

 equation (80) I am indebted to Mr. G, H. HARPY, Fellow of Trinity College, Cambridge. 



2 c 2 



