Bauhaus-Universität Weimar

Titel:
On sound and atmospheric vibrations, with the mathematical elements of music
Person:
Airy, George Biddell
PURL:
https://digitalesammlungen.uni-weimar.de/viewer/image/lit38525/92/
76 
ON SOUND. 
with the four values for m, 1, 2, 3, 4. It does not 
appear that equations of this class can be approached by 
one general method of attack. Some of them yield to 
the following. Assume, for trial, W= 2 (An. rn) ; A 
being in all cases a function of v or at — r, which satis- 
1 d2A d2A 
fies the equation ; and the index n 
diminishing by successive units. On substituting, and 
remarking that A! or ^ ~ , we find 
dv dr 
2 {— (2n + m). Än. rn~x 
+ n (m + n — 1). An. rn~2} = 0, 
(the accent on An denoting the derived function of An 
with regard to v) ; 
and, writing down successive terms instead of the sym¬ 
bol 2, 
{— (2n + m) . A'nj . rn~l 
+ {— (2n + m — 2). n (m + n — 1). Anj. rn~2 
+ {— (2n+m — 4).A'n_2-h(n—l)(m+n — 2).An_1}.rn~3 
&c. 
[If we suppose A a function of u or at + r, the 
equation obtained is the same, excepting that the signs 
of the derived functions are changed]. 
Making each line = 0, we find n = — ~f and we find 
each following function in terms of the preceding func-
        

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