674 DR J. HALM 
two simple transcendents just now mentioned. It is, indeed, easily seen that the values 
of G(c, 1) and G(c, 1) are approximately represented by 
a 
2'eos[ Ve + ylog (1+ V2)] and sin [Ve + y log (1 + N2)] 
2 
J c+y 
where y is a certain constant. 
By slightly altering the constants, these two expressions can be made to represent 
the - and ©-functions even more closely. We have then approximately 
G(c, 1) = 1719942 x cos [r/c + 0°438 x 50°458] 
Je+0:438G(c, 1) =1:19942 x sin [Vc + 0°438 x 50°°458]. 
If we calculate the expressions on the right-hand side for the values of c given in 
Table III., by comparison of the calculated figures with the exact values of @(c,1) and 
©(c, 1) shown in the same table, we find the corrections AG and AS which have to be 
added to the computed transcendents. These corrections are conveniently small to 
permit an easy interpolation, and we are thus in a position to ascertain the correct 
numerical values of G(c, 1) and G(c, 1) for values of c lying between those mentioned 
in Table III. by simply computing the corresponding values of the right-hand sides of 
the above approximate equations and by adding the corresponding corrections A@ and 
AG interpolated from the figures of the same table. In this way the following Table IY. 
has been constructed, which shows, for conveniently small intervals, the @(c,1) and 
S(c’, 1) for the arguments ¢ between zero aud 30°0. To exhibit at the same time, more 
concretely, the character of the functions C(c,1), S(c, 1), E(c, 1)and S(c, 1), their 
graphs are given in fig. 2. 
(Pareiam IW 
oa Cc, 1) | Slr, 1) c C(c, 1) Ge, 1) 
0:0 + 1:0000 + 1:0000 13°0 — 1:1946 — 0:0259 
0:5 +0°7877 + 0°9353 i 14:0 = Welt — 0:0619 
1:0 + 0'°5894 +0°8731 15:0 — 11389 — 0:0932 | 
15 + 0°4047 + 0°8133 16:0 — 1:0908 — 0:1208 
2°0 + 0:2328 +0°7557 17:0 — 1:0314 —0'1446 
2°5 + 0:0730 + 0:°7008 18:0 — 0:9626 — 01648 
3°0 —0:0748 + 0°6480 19:0 — 0°8854 - = Oorsiny 
Fa) = (OPAL + 0°5974 20°0 —0°8019 — 01956 
4:0 — 0°3368 + 6°5489 21:0 — 0'7126 — 0:2067 
4:5 — 0°4519 + 0°5024 22:0 — 0°6192 — 0°2153 
5:0 = 05568 + 0°4580. 23°0 — 0°5224 — 0:2214 
6:0 — 0°7388 +0:°3747 24.0 — 0:4235 — 0'2255 
7:0 — 0°8856 + 0°2989 25°0 — 0:3229 = OR2275) 
8:0 — 1:0007 + 0'°2299 26:0 — 0:2225 S(T 
9:0 — 1:0868 +0:1673 27:0 —0:1219 — 0:2264 
10:0 — 11468 +0°1108 28:0 — 0°0226 =I) 2204 
11:0 = elie +0:0600 29:0 + 0:0755 — 0'2192 
120 = 984 +0:0146 30:0 = Oa iOr ==) 11 33/7 
