192 PROFESSOR A. E. H. LOVE ON THE 



must hold in order that the frequency equation may be satisfied by p 2 0. When p* 

 vanishes, h 2 and & 2 also vanish, but the quantity tfjh 2 , which is (X+2/i)//ii, has 

 a determinate value. We may not, however, obtain the result which we seek by 

 first replacing P//t 3 , wherever it occurs, by (X + 2//, )///., and then putting h? and A.- 2 

 equal to zero wherever they occur, otherwise than in the ratio Ifjh 3 . This pre- 

 cautionary statement is necessary because it appears from the formulae (69) and (70) 

 of 17 that C,, and D B both vanish if h' and P vanish. Thus we ought to regard 

 the equations (71) as being equivalent to the equations 



in other words, we ought to remove a factor F from the equation A B D B B B C B = 

 before putting h 2 = and k? = 0. An exceptional case occurs when n = 1. In this 

 case C B yt~ 2 vanishes when h 2 vanishes, and it will appear that A B also vanishes with h 2 , 

 and the equations (71) ought to be regarded as equivalent to 



= 0, B^ + D^- 2 (jfcty) = 0, 



and we must remove a factor hfk 2 from the equation AiDj B^ = before putting 

 It 2 = and k 2 = 0. . When we proceed in this way, the equation A B D B B B C B = 0, 

 with the appropriate factors removed, and with h 2 and k 2 put equal to zero after 

 their removal, becomes an equation to determine 2 a 2 , or |7ry/3 2 a 2 /(X + 2/i). If the 

 equation has a real root', the value so determined for s 2 a 2 gives a value of X+2/z. for 

 which instability can occur. Since P<f> n is a finite multiple of <a n when X+2/n has any 

 such value, it is certain that the homogeneous spherical configuration really is unstable 

 for such values of X+2/A. If the value of X+2/4 belonging to the body is but little 

 greater than the critical value, the equilibrium is practically unstable ; for a large 

 displacement takes place if the sphere begins to vibrate according to the type 

 specified by the degree n of the corresponding spherical harmonic function. For 

 practical stability it is necessary that the value of X+2/u, should be well above any 

 critical value. The equation which yields the critical values contains the constant 

 (X+2/u.)//Lt as well as sV. It will be convenient to write 



v = 



x+^'jf ;-' ; ' ; * (73) 



The value of v cannot be negative, nor can it be greater than f . If the POISSON'S 

 ratio (X/2(X+/it)} of the material is positive, v cannot exceed \. If the modulus of 

 rigidity \L were very small in comparison with the modulus of compression X+/u, 

 v would be very small. If the velocity of propagation of waves of dilatation were 

 twice that of waves of distortion, v would be ^. This appears to be the most 

 appropriate value to assume in the case of the Earth (see 40 below). Since it is 

 improbable that the ratio of the rigidity to the modulus of compression of the Earth 

 has diminished since the date of consolidation, it will be sufficient for our purpose to 



