458 
PROFESSOR STOKES ON THE COMMUNICATION OE VIBRATION 
for U will introduce a function ->!/ n of the same order in the general expression for <p. 
Now the only case of interest relating to an infinite cylinder is that of a vibrating string, 
in which the cylinder moves as a whole. The vibration may be regarded as compounded 
of the vibrations in any two rectangular planes passing through the axis, the phases of 
the component vibrations, it may be, being different. These component vibrations may 
be treated separately, and thus it will suffice to suppose the vibration confined to one 
plane, which we may take to be that from which Q is measured. We shall accordingly 
have 
U=Uj cos 0, 
U, being a given constant, and the only function which will appear in the general 
expression for <p will be that of the order 1. Besides this we shall have to investigate, 
for the sake of comparison, an ideal vibration in which the cylinder alternately contracts 
and expands in all directions alike, and for which accordingly U is a constant U 0 . Hence 
the equation (15) need only be considered for the values 0 and 1 of n. 
For general values of n the equation (15) is easily integrated in the form of infinite 
series according to ascending powers of r. The result is 
+ 2 n)' 2.4 (2 + 2n)(4 + 2ri) 
-■}] 
2(2 — 2 n)~ 2.4(2 — 2w)(4— 2n) 
• • (16) 
When n is any integer the integral as it stands becomes illusory ; but the complete 
integral, which in this case assumes a special form, is readily obtained as a limiting case 
of the complete integral for general values of n. 
The series in (16) are convergent for any value of r however great, but they give us 
no information of what becomes of the functions for very large values of r. 
When r is very large, the equation (15) becomes approximately 
^+m*4-.=0, 
the integral of wdiich is •*]/„= Be H'e imr , where B and B' are constant. This suggests 
putting the complete integral of (15) under the same form, B and B' being now functions 
of r, which, when r is large, vary but slowly, that is, remain nearly constant when r is 
altered by only a small multiple of X. Assuming for B and B' series of the form 
Ar a + B/^+Cr Y . . ., where a, (3, y .. . are in decreasing order algebraically, and deter- 
mining the indices and coefficients so as to satisfy (15), we get for another form of the 
complete integral 
4,=C(imr)->e-< mr h-^ 
( 1 2 — 4re 2 ) (3 2 — 4n 2 ) 
1 .2(8 imr) 2 
(l 2 — 4ra 2 ) (3 2 — 4n 2 ) (5 2 — 4n 2 ) 
1.2. 3 (8m) 3 
-f-D (imrY^e" 
(l 2 — 4n 2 ) ( 1 2 — 4ra 2 ) (3 2 — 4n 2 ) (l 2 -4rc 2 )(3 2 -4n 2 )(5 2 -4rc 2 ) 
1.8 imr l.2(8mr) 2 * i » + 
1.2.3(8w) 5 
•I 
( 17 ) 
