178 PROFESSOR A. E. H. LOVE ON THE 



5. We shall now suppose that the system executes a normal, or principal, vibration 

 of frequency p('2ir, or, in other words, that every component of displacement is 

 proportional to the same simple harmonic function of pt. The equations of vibratory 

 motion become three equations of the type 



0, .... (15) 



where W satisfies the equation (11). The solutions of this system of equations (11) 

 and (15) must be adjusted to satisfy the conditions (13) and (14) at r = a. These 

 conditions can be satisfied only if p has one or other of a certain infinite set of values, 

 which are the roots of the frequency equation. The problem of gravitational 

 instability is solved when we find the conditions that one of the values of p may be 

 zero. 



Solution of the Differential Equations by Means of Sphencal Harmonics. 

 6. We introduce the notation 



The equations of motion (15) become three equations of the type 



x 

 and the equation v 2 W = 47ry/3 A becomes 



V 3 E = A; (18) 



in these equations A stands for 



dx dy dz ' 



By differentiating the left-hand members of the equations of type (17) with respect 

 to x, y, z, respectively, adding the results, and simplifying by means of (18), we 

 obtain the equation 



=0 ......... (19) 



The method of solution of the problem is this : We seek first a solution of the 

 equation (19) in which A has the form 



, .......... (20) 



where ta n is a spherical solid harmonic of positive integral degree n, and / is a 



