380 
PROF. LOUIS VESSOT KING ON THE CONVECTION OF 
Integrating term by term the asymptotic expansion (9), we obtain 
I e u K 0 (w) du +1 = y/( 2ttx) 
Jo 
1 + 
l 2 .3 
+ 
l 2 . 3 2 .5 
8cr (8x ) 2 .2 ! (8x) 3 .3! 
• • (24) 
The term 1 on the left-hand side represents a term of integration arising from the 
lower limit of the integral, which is proved by actual numerical calculation to 
be unity. 
In order to evaluate F ( x ) numerically over the interval where neither (23) or (24) 
are convergent, use is made of Euler’s formula for the quadrature of the function 
y =/(«)( 10 ) 
j ydx = h (ly 0 + y 1 + y 2 +---h/ n ) ~ i\h 2 If M ~f M] 
+ M-f" {x 0 )]-..., . . . (25) 
where h is the interval between the successive values of x, that is, h — (x n —x 0 )/n. 
From a table ( u ) of the Bessel’s functions K 0 ( x ) and Kj (x), the values of e^Ko ( x) 
and e x Ki (x) were tabulated over various ranges of equidistant intervals from x = 0T 
to x = 6‘0. Numerical values of F (x) for values of sc'<0T were easily calculated 
from the convergent formula (23). Beyond this point the integral between the 
limits x = O’l and x — x n was calculated by the use of Euler’s formula (25), making 
use of a calculating machine for the purpose in such a way that the various entries 
were recorded as successive values of a single series of operations. Beyond x = 6'0 
the function was evaluated from (24), the constant of integration proving to be unity. 
The functions ( x ), e x YL 1 (x), j e z K 0 (x) dx and xj J (x) dx are tabulated in 
Table I. together with a more detailed description of the method of computation ; 
a graph of the last function is also given. 
Section 6. Approximate Formula for the Heat-Loss. 
When the variable x is small, equation (23) enables us to write 
y = x 
e x K 0 (x) dx 
in the form y = 1 /[(1 — y)— log \x\. 
(26) 
Hence when the variable n/3 0 is so small that it may be neglected in comparison with 
log (l/n/3 0 ) the expression (21) for the heat-loss may be written in the form 
H = 2^0 Q /[log(46/)8 o )],. (27) 
where 
c = so- and b = e 1 ~ y /(2n) = /te 1 y /(sa-Y). 
(28) 
( 10 ) Euler, ‘Comm. Acad. Sci. Imp. Petrop.,’ vi. (1732-33). 
( n ) Jahnke and Emde, ‘ Funktionentafeln ’ (Teubner’s, 1909), p. 135. The functions (in/2) Ho 1 («) 
and - (7 t/2)Hi 1 (ix) of the above tables are here denoted by K 0 (x) and Kj (a) respectively, and their 
computation is due to W. S. Aldis (‘Roy. Soc. Proc.,’ vol. 64, p. 219, 1898). 
