460 
PROFESSOR STOKES ON THE COMMUNICATION OF VIBRATION 
a formula of reduction which when n is integral serves to express 4 n in terms of Mo- 
have 
We 
( 21 ) 
an equation which when applied to (18) gives the complete integral of (15) for integral 
values of n in the form of a definite integral. 
Let us attend now more particularly to the case of n= 0. The equation (16) is of 
the form ^ n —Af(n)+¥>f(—n),f(n) containing r as well as n. Expanding by Maclauein’s 
Theorem, we have 
^=(a+bmo)+(a-b}/’(o>+(a+b)/"(0) i A+ . . . 
Writing A for A-(-B, n ‘B for A— B, and then making n vanish, we have 
4-.=A/(0)+B/(0), 
or 
i / a i -d i \f-t m2/l2 , m4/ ' 4 
^ 0 =(A+Blogr)( 1— . . • 
+ B 
S n =l“ 1 -l-2- 1 + 3- 1 ... +W" 1 . 
m 2 r 2 _ m 4 r 4 „ m 6 r 6 \ 
22 2 2 4 2 ■ 2 2 4 2 6 2 • • • J ' 
( 22 ) 
where 
The integral in the form (17) assumes no peculiar shape when n is integral, and we have 
at once 
%|/ 0 = C(imr)~ i e~ imr - 
+D (mr)~V mr j 
I have explained at length the mode of dealing with such functions, and especially of 
connecting the arbitrary constants in the ascending and descending series, in two papers 
published m the Transactions of the Cambridge Philosophical Society*, in the second 
of which the connexion of the constants is worked out in this very example. To thfese 
I will refer, merely observing that while it is perfectly easy to connect A, B with E, F, 
the connexion of C, D with E, F involves some extremely curious points of analysis. 
The result of eliminating E, F between the two equations connecting A, B with E, F 
and the two connecting C, D with E, F is given, except as to notation, in the two equa- 
tions (41) of my second paper. To render the notation identical with that of the former 
paper, it will be sufficient to write A— Blog(m)+Blog(mr) for A+Blogr, and x for 
imr. The equations referred to may be simplified by the introduction of Euler’s con- 
stant y, the value of which is -57721566 &c., since it is known that 
1 - 
1 2 3 2 
1 2 3 2 5 2 
1 + l| 
1 . 8 mr 
l 2 
1 . 2(8 mr)~ 1.2. 3(8 »r) ! 
1 2 3 2 1 2 3 2 5 2 
+ ... 
8 imr' 1 . 2(8 imr) 2 1.2. 3(8mr) £ 
+ 
■ (23) 
5r ~*r'(i)+log4+y=0, 
* “ On the Numerical Calculation of a Class of Definite Integrals and Infinite Series,” vol. ix. p. 166, and 
“ On the Discontinuity of Arbitrary Constants which appear in Divergent Developments,” vol. x. p. 105. A 
supplement to the latter paper has recently been read before the Cambridge Philosophical Society. 
