AND THE ACTION OF TWO VORTICES IN A PERFECT FLUID. 
519 
Let us consider the more general integral 
cos scf)d(f) 
u— \ , c, . , 0 .2p + l 
0 
Considered as a function of s it is easy to prove that the integral satisfies the 
differential equation 
dhi 2\p — 1 du 
ds 3 2s ds 
or if u=sPv 
■ahi—0 
d % v , 1 dv Ar , o 
7 o+~ ~y~ v \ 
ds- 1 s ds \s z 
0 
The solution of this differential equation may be expressed in terms of Bessel's 
functions of the first and second kinds • we shall, however, only solve here the special 
case which we require. 
It is easily proved that 
f 06 cos s(f) 7 , s d cos scf) 1 x 
■ J 0 (a 3 + f‘) ,< *’ > " _ ~a 3 &J 0 (VU=y- 'V 
Since p —0 for 
Jo 
COS S(j) 
~ (cd + ^ satisfies the differential equation 
dhi 1 du 
ds 3 s ds 
—ahc= 0 
This is the equation solved by Professor Stokes in his paper “ On the effect of 
Internal Friction on the Motion of Pendulums,'” Camb. Phil. Trans., 1850, and quoted 
by Sir William Thomson, Phil. Mag., September, 1880. 
The solution is there shown to be 
cds 2 . 
«=(E+DlogfJ(l+^+^+ 
as 
f a~s 9 - 
~hhl( 22 ^1 + 92/i2^3“h • • • 
2 3 4 2 
where 
S,— 1 1_ b2 1- f- . . . n - 1 
and if as for 
COS S(f) 
(a a + ^ 8 ) 4 ^, u= ® w ^ en s 4S infinite E/D is shown to be 
mdccclxxxit. 
log 8+ir~*r , i=T1593 
8 x 
