GRAVITATIONAL STABILITY OF THE EARTH. 201 



28. In the case where n = 1 the condition of gravitational instability, viz., 



- 1 Jfc' il ) = 0, becomes 



_Att ^ cf I A/ 2 _ 1\ ,*, 2A, ^-.r^ e - k * dx \ = o (88) 



105V 8?15\ vl 5i> v Jo 



But we have 



and therefore the condition of gravitational instability becomes 



3( m e-^dx-e-^\38a+(^aY + (^-- v }(m) & } = 0. 

 Jo I \2l) / 



If now we put 



= x 9 = 2z', 

 the equation becomes 



7rJ 



=0, . . . (89) 



where the factor 2ir~* has been inserted because the expression in the square brackets 

 is tabulated in many easily accessible books. 



Let y denote the left-hand member of the equation (89). When z is small, y is 

 small of order z & . In fact, we have 



and when z is small, the first approximation to y is (3 4i>)z & . When z is great, 



I e'^dt is approximately equal to v/ir ; and thus y is positive when 2 is great. The 

 Jo 



equation (89) has a root zero and at least one positive root. The zero root is 

 irrelevant to our problem ; it is introduced in transforming equation (88) into 

 equation (89). Now we have 



^ = -zV 1(15-20,,) -|(19-20i/)z f l ; 



and, since 15 20^ and 19 20^ are positive when v<f, the expression last written 

 vanishes for one positive value of z. Hence it follows that the equation (89) has only 

 one positive root, and there is one and only one positive value of **a* which satisfies 

 equation (88). 



By means of the tables it can be shown that, when v = 0, the root z lies between 

 1*9 and 2, so that sV lies between 7 '22 and 8. When v = , the root z lies between 

 1-8 and 1'9, so that sV lies between 6'48 and 7'22. 



VOL. ccvir. A. 2 D 



