410 
PROF. LOUIS YESSOT KING ON THE CONVECTION OF 
Table of the Functions e x Ko (*), 
e^Ki 
e x K 0 (x) dx, 
•/[ f; 
e*K 0 (x) dx 
The functions tabulated in the first two columns of the table opposite are based on Jahnke and Emde’s 
tables( J ) of the functions K 0 {x) and Ki (z) for values of x between x = O' 1 and x = 12 - 0 The process of 
multiplication by the factor e x was easily carried out by means of a “ Brunsviga ” calculating machine.( 2 ) 
For values of the argument x < O’l the functions were calculated directly from the convergent 
expansions 
K 0 (»') — - Iq (*) [y + log 4-' ; ] + ^ + 9 2 -^o ( 1 + i) 
- Kj (x) = Ij (a:) [y + log 4x] + 
1 a:/2\ 2 lx 
2 2 .4 2 .6 2 
(1+4 + 4)+. 
o o 
\ x: 
o o 
\_ ; 1 + 1+4 /rrV , 1 +4+ 1 +4 + ^ (xf 
2 !. 3 ! (2) 
1 ! 1 !. 2 ! \ 2 
+ ... 
. (i.) 
. (ii.) 
For large values of the argument numerical values were obtained from the asymptotic expansions 
«*k.« = g h 
r 1 _ 1 2 .3 2 
■ ea ' Ki (x) = ( — 
8:r 2 ! (8.r)‘ 2 
i + 2_^_ 5 „ + 3 - 5 - 7 
1 ■ / r-v \ O 1 
• (m-) 
• (iv.) 
8x 2 ! (8a;) 2 3 ! (8a;) 3 
For values of x< 0*1, the function j e^K,, (x) is most easily calculated from the convergent series 
J o 
obtained by integrating (i.) term by term after multiplying by the factor e x : 
| (( *K. (x) dx = 1 + ia; + \x l + g\x 3 + ... - (y + log \x) (1 + \x + +r 2 + gx 3 +...). . . (v.) 
From the point x — O' 1 onwards it is convenient to make use of Euler’s ( 3 ) formula for quadratures in 
the form 
2/r 1 [ ydx = [(y 0 + ?/i) + (yi + y 2 ) + ■.. + (y n -i + Vn)\ - [f (x n ) -f (z 0 )] + ^\gi 3 [/"' (x„) (a; 0 )] - ... (vi.) 
J Xo 
In the case under consideration we have 
T (x) = e* [K 0 (x) + Kj (*)], /"' (a;) = (4 - 3ar' + 2a;- 2 ) e*K T (x) + (4 - xg e*K 0 (x). . . (vii.) 
The first term on the right-hand side of (vi.) in square brackets [ ] is especially well suited for 
computation on an adding machine; the totals (yo + yi), (yo+f/i) + (?/i + //?), &c., are obtained as successive 
results of a single series of operations and are thus easily tabulated. Opposite these entries are written 
down the values of ~hf (x n ), obtained by using equation (vii.) from the first two columns of Table I. The 
third order correction term in (vi.) can be easily shown to be too small to affect the value of the series to 
nn 
1 part m 1000, as seen from the numerical example given below. Starting with ydx = O'3572 
calculated from the convergent series, and taking h = 0* 1, the work is tabulated as follows:— 
X. 
y . 
W (*)■ 
f 01 
2 h 2 ydx + S„. 
C'n. 
n* 
fX u 
y dx. 
Jo 
0-1 
2-6824 
-0-1368 
-0-0052 
— 
— 
— 
0-3572 
0-2 
2-1407 
-0-0615 
-0-0006 
11-9681 
0-0753 
0-0046 
0-5946 
(‘) See footnote ( n ). 
( 2 ) For a description of this instrument see d’Ocagxe, ‘ Le Calcul Simplifie ’ (Gauthier Vi liars, Paris, 1905), pp. 62-65. 
( 3 ) See footnote ( 10 ) 
