MR. J. H. JEANS OX THE VIBRATIONS AND 
ir>2 
artificial case appears to be the only case in which the equations can be solved by 
ordinary analysis. 
We now replace P, Q, R, S, T, U by their ordinarily assumed values, and ecjuation 
(7), putting W = 0 , takes the form 
P fjf. — (^ + d) 0 .. + 
0E 
• • (8), 
and there are two similar equations for 77 , 
The P rinci'paJ Vihratiom and Frequencij Eqvritions. 
9. Difterentlate these three equations of motion with respect to x, y, z and add; 
then 
p ^ = (X + 2p) V^A - pV^E.(9). 
Now, from the definition of E, we have, in the case in which p is constant, 
= — 47rpA.(10), 
and hence equation (9) becomes 
P = (k + 2p,) WA + Jvrp^A.(11). 
If we suppose A proportional to cos ]>f, this equation assumes the form 
(V^ fi- «:“) A = 0, where 
_ p(p“ + Jtt/ j) 
X -|- 2y£X 
( 12 ). 
There is, therefore, a particular solution of ( 11 ) of the form 
A = r“^J„^.j(«:r)S«(d, (f)) cos pt .(13), 
vheie cf)) is a surface harmonic of order n, and the general solution found bv 
summation of solutions of this type is 
A =--^'“-J;, + j(/cr) (^?, (^) (A cos 7 V + B sin .... (14), 
wheie the summation extends over all possible harmonics, and over alb values of k. 
It V ill appeal later that each term in this solution can be made to satisfy the 
boiindaiy conditions, and, therefore, that each term represents a normal vibration. 
The vibrations may, therefore, be classified into vibrations of order 0 , 1 , 2 , &c., the 
order being that of the harmonic which occurs in the expression for A. The vibrations 
of order n = 0 are the spherical vihrations already referred to. 
