HEAT FROM SMALL CYLINDERS IN A STREAM OF FLUID. 
377 
of which a particular solution appropriate to a line source at the origin is seen, on 
transforming ( 6 ) to cylindrical co-ordinates, to be ( 7 ) 
cf> = AK 0 (nr), where r = | \/(x 2 + y 2 ) [, x = r cos 6, y = r sin 0 , . (7) 
A is a constant of integration and K 0 (z) that solution of Bessel’s equation most 
conveniently defined by the definite integral 
( 8 ) 
We write down for future reference the expansions 
K 0 (z) = e V^/ 2 *:) 
' 1 , l 2 -3 2 
8z ( 8 z) 2 . 2 ! 
..(9) 
when 2 is large. When z is small we may make use of the expansions 
+ (io) 
K 0 (z) - I 0 (z) [y + log (z/ 2 )]+ z — + 2 ^-^(l + 2 ) + 02 
where 
I 0 ( 2 ) = 7 T - 1 | cosh (2 cos (p ) d<p = l+^ + 2 2 >4 2 ■ 2 2 .4 2 .6 2 
z 2 Z* 2 6 
+ ..., . . (II) 
and y is Euler’s constant, y = 0'57721. 
If we denote by Q the rate at which heat is being supplied to the line source per 
unit length, Q should also be equal to the total flux calculated by integrating around 
any closed circuit enclosing the source; that is, 
Q = 
— K (do/dv) +cV cos 
. 0 ] ds, 
(12) 
where the integral is taken around any closed circuit c enclosing the source, and 
V cos e is the component of velocity along the outward drawn normal v. This 
remarkable property of the solution 6 = Ae njr K 0 (nr) is easily verified by carrying out 
the integration around a circle of radius r, the result serving to determine the 
constant A. Remembering that 
dK " ( z) = K x (z) and = .( 13 ) 
dz az 
we have dOjdr = Ane nrcos6 [K x (nr) + cos 6 K 0 (nr)\ and thus from ( 12 ) 
Q = 2 r j [271k 6 cos e—K(dd/d7’)]de = 2Au<nr —K x (nr)| e nrcoa * de + K 0 ( nr )j^cos ee* TC0 *'de . 
( 7 ) Gray and Matthews, ‘ Treatise on Bessel Functions,’ 1895, pp. 77 and 90. The notation employed 
throughout is that of the above treatise. 
3 C 
VOL. CCXIV.-A. 
