210 PROFESSOR A. K H. LOVE ON THE 



Since 



' *-'* 1 dx, 



o 



the expression (101), when multiplied by 7, is 



(5x'-36x a + 1 54) \*a* e -** dx+5(a*-S)a*e-** ..... (102) 



Also we have 



FX 



*dx = 



f 



Jo 



9 9. 11 9.11.13 "/' 



and therefore the coefficient of x 9 e~^ in the expansion of (102) is (154-9 x 15), the 

 coefficient of xW is ^(154-36x11 + 5x99), and the coefficient of x 9+3x e,-** for 

 all values of K which are greater than 1 is 



36 4J), 



9. ll...(2/c+5)\(2/f+7)(2/c+9) 

 or 



20^+88^+145 

 9. ll...(2fc+5)' 



Hence all the coefficients are positive, the expression (102) is positive, and the left- 

 hand member of (100) is positive for all positive values of x. Thus, in this case also, 

 the equation (99) has no real root. 



Summary of the Solution of the Mathematical Problem. 



34. We have now solved in essentials the mathematical problem of the vibra- 

 tions of a gravitating sphere, initially homogeneous and in a state of hydrostatic 

 pressure, and have found the conditions of gravitational instability. We have shown 

 that, when any normal, or principal, vibration is taking place, the dilatation at 

 a distance r from the centre is specified by the product of a certain function of r and 

 a spherical harmonic of positive integral degree. We have shown further that, in 

 each such mode of vibration, the components of displacement can be expressed in 

 terms of the same spherical harmonic, and that the radial displacement at a point 

 distant r from the centre is the product of a function of r and the same harmonic. 

 We obtained the form of the frequency equation, and the forms of all the functions 

 which enter into its expression. 



We proceeded to investigate the conditions which must hold in order that the 

 frequency equation may be satisfied by a zero value of the frequency. We showed 

 that, when such a value is not introduced irrelevantly in the process of forming the 

 equation, its occurrence points to genuine gravitational instability. We found that 

 the condition of such instability is the condition that a certain equation, containing 

 the variable quantity x'a*, or f7ry/j u 2 /(X + 2/i), may be satisfied by a real positive value 



