of Edinburgh, Session 1878 - 79 . 
99 
are the speeds * * * § of the successive fundamental modes, corresponding 
to the different circular nodal subdivisions of the i diametral divi- 
sions implied by the assumed value of i. Thus, by giving to i the 
successive values 0, 1, 2, 3, &c., and solving the transcendental 
equation so found for each, we find all the fundamental modes of 
vibration of the mass of matter in the supposed circumstances. 
If there is no central island, the solution of (19) which must be 
taken, is that for which P and its differential coefficients are all 
finite when r = 0. Hence P is what is called a Bessel’s function of 
the first kind and of order i ; and according to the established 
notation t we have 
-’W&) 
( 21 ) 
The solution found above for an endless circular canal is fallen 
upon by giving a very great value to i. 
Thus, if we put = A. so 
that \ may denote wave-length, we have ~ = which will now be 
the m of former notation. We must now neglect the term - $ 
° r dr 
in (19), and thus the differential equation becomes 
- 4 G > 2 
dfk 
dr 2 + 
gv 
2 U = o 
d 2 h 
dfi-n-o 
where l 2 denotes m 2 - 
h = ce ~ } y where y = a- 
4o> 2 
. . . ( 22 ), 
A solution of this equation is 
~9 
r, and using this in (20) above, we find 
£ = ^2 _ 4^2 C sin (mx - at) (at - 2 where mx — iO. Hence, 
to make £ = 0 at each boundary, we have al = 2 wm, which makes 
* In the last two or three tidal reports of the British Association the word 
“speed,” in reference to a simple harmonic function, has been used to 
designate the angular velocity of a body moving in a circle in the same period. 
2 ?r 27 r 
Thus, if T be the period -jp is the speed ; vice versa, if a be the speed — is 
the period. 
t Neumann, “ Theorie der Bessel’schen Functionen” (Leipzig, 1867), § 5; 
and Lommel, “ Studien iiber die Bessel’schen Functionem ” (Leipzig 1868), 
§ 29. 
