HEAT FROM SMALL CYLINDERS IN A STREAM OF FLUID. 
411 
where 
S n = (?/o + 2/1) + (?/i + 7/2) + {y-i + 2/3) + • • • + (t/ji-i + ?/»)> 
G'„ = fr[f(z n )-f(x 0 )], 
and 
C" 
1 
nnr' 
By carrying out this process the function | e x K 0 (x) do: was tabulated as far as x = 6-0, beyond which 
point the calculation by means of the asymptotic series involves less labour. We obtained, in equation (24), 
an expansion of the form 
f 
e*K 0 (x) do: 4 - C 
s /(2wr) 
1 + 
So: 
l 2 .3 1 2 .3 2 .5 
2 ! (8a:)" 2 + 3 ! (8 x) 3 
(viii.) 
That the constant C is unity is very probably true although an analytical proof of the result would be 
very laborious; for the purposes of the present work the case is sufficiently justified by the following 
numerical calculations of the sum of the series on the right-hand side of (viii.). These calculations, at the 
same time, furnish a check on the results obtained by quadratures :— 
x = 1. 
Series (v.) ... 
. . 1-779 
Series (viii.) . . . 
• • [2-778]e 
Quadratures . 
. . 1-7804 
C. 
. . [1-002] 
x = 2. 
x = 3. 
x — 6. 
3-7519 
4-5165 
6-2643 
2-7499 
3-5127 
5-2640 
1-0020 
1-0038 
1-0003 
The values of the function 
xj jj e x K 0 (x) dx 
are easily obtained from the functions previously 
tabulated. This function when plotted against Jx, as shown in Diagram I., gives very approximately a 
straight-line relation over the range dealt with in the present experiments. Simple approximate formulae 
for the function in question are derived and tested in the description of Diagram I. 
( 4 ) The series (viii.) fails to give accurate values for so small a value of the argument as r = 1. 
2 ‘798, and of five 2 "758, the value given above being the mean of these two determinations. 
The sum of four terms is 
3 G 2 
