ON LINEAR DIFFERENTIAL EQUATIONS OF THE SECOND ORDER. 671 
interpolation of these functions for intermediate values of c. If required, the table 
may be extended by means of Table I. and the preceding formule. 
Tasue II. 
c Cie, 1) S(c, 1) C Ce, 1) S(e, 1) 
0:0 + 1:00000 + 1:00000 5:6 — 1°50392 + 0:02851 
0:2 + 0°86568 + 0:94030 5:8 — 0°50292 +0:01375 
0:4 + 0°74008 + 6°88215 6:0 — 050000 0:00000 
0°6 + 0°62247 + 0°82674 7:0 — 0°46052 —0:°05615 
0:8 + 0°51257 + 0°77390 8:0 - 0'38976 — 0:09397 
1:0 + 0:41034 + 0°72302 9-0 — 0:°29986 — 011695 
iy + 0°31593 + 0°67440 100 — 0:20033 — 0'12809 
1-4 + 0°22694 +0°62778 11:0 — 0:09837 — 0:13006 
1°6 +0:14511 + 0°58331 12:0 0:00000 — 0°12500 
1:8 + 0:06956 +0°54071 13:0 + 0:09082 —0:11479 
2:0 0:00000 + 0°50000 14:0 +0:17106 -—0:10101 
eo) — 0:06385 +0:°46111 15:0 + 0°23s02 — 0:08524 
2°4 — 0712221 + 0°42406 16.0 + 0°29323 — 0:06753 
2°6 —0:17561 | + 0°38864 17.0 + 0°33377 — 0:04971 
2°8 — 0°22395 + 0°35492 18-0 + 0°36035 — 0:03222 
3:0 — 0'26758 + 0°32282 19:0 + 0:37382 —0'01550 
3°2 — 0°30673 + 0°29227 20°0 + 0°37500 0:00000 
3°4 —0°34177 + 0:26317 21:0 + 0°36508 + 0:01404 
3°6 = OSV ONS + 0:23555 220 + 0°34531 + 0°02640 
3°8 — 0°39987 + 0°20933 23°0 + 0°31710 + 0°03698 
4:0 — 0:42343 + 0°18442 24:0 + 0°28192 + 0:04572 
4°2 -0°44360 + 0°16083 25°0 + 0°24103 + 0:05266 
4°4 — 0°46051 + 0°13853 26:0 +0:19611 +0°05779 
4°6 — 0°47435 +0°11741 27°0 + 0714830 + 0:06123 
4°8 — 0°48542 + 0:09744 28°0 + 0:09889 + 0 06308 
5:0 — 0°49366 + 0:07865 29°0 +0:04919 + 0:06345 
52 —0°49941 + 0'06088 30°0 0:00000 + 0:06250 
5:4 - 0:°50277 + 0°04422 
As regards the hyperbolic functions @(c , 1) and G(c, 1), it seems, however, impossible 
to calculate their values by means of the corresponding relations, because the 
P-functions, which appear in the expressions, have imaginary arguments. But 
fortunately, in this case, the series are far more manageable than those of the C- and 
S-functions, so that the labour involved in their direct calculation is not nearly so 
stupendous as it may appear at first sight, even if an accuracy within the fifth decimal 
place is desired. To show this let us first consider the series 
jeeee atl i (e+1)(c+9) (c+ 1)(c+9)(c + 25) 
2.3 2.3.4.5 2.3.4.5.6.7 
which, as we know from the preceding investigations (see (29) ), is rigorously represented 
Geeta rs. . Lear 
by the transcendent We sin (,/¢ log (1 + ./2))- We want to inquire how many terms 
of the series are necessary to obtain the accurate value of this transcendent within, say, 
five places after the decimal point. If we call =(x) the sum of the x first terms, and if 
