ON LINEAR DIFFERENTIAL EQUATIONS OF THE SECOND ORDER. 673 
decimal place. The same number of terms is therefore also sufficient for the calculation 
of the more convergent Seiche-function 
pha nes. Ae Oe) 
ure 231934.) 2.3.4.5.6.7 
for the same value of c. This method enables us to compute with sufficient accuracy 
the S- and @-functions, even for great values of the argument c, from a comparatively 
limited number of terms of the series. I subjoin the results of my computations in the 
following Table III. : 
TABLE Wl: 
¢ (S(c, 1) Sie, 1} AC AS 
0 + 1:0000 + 1:0000 — 0:0014 + 0°0025 
1 + 0°5894 +0°8731 — 0:0010 + 0:0024 
2 + 0°23258 +0°7558 — 0:0006 +0°0025 
3 — 0:0748 + 0°6479 — 0:0003 +0:0023 
4 — 0°3368 +0°5490 — 0:0002 +0°0025 
5 — 0°5568 -+ 0°4580 0:0000 + 0:0024 
6 —0'°7388 +0°3748 + 0:0001 +0:0025 
8 — 1:0007 +0°2299 +0°0003 +0 0024 
10 — 11468 +0:1108 +0°0003 +0:0024 
12 — 1:1985 + ():0147 +0°0002 + 0:0025 
16 — 1:0908 —0°1208 0-0000 + 0:0022 
20 —0°8019 —0'1956 —0:0010 +0 0019 
24 — 0°4235 — 0°2255 — 00021 +0:0017 
28 — 0°0226 — 0°2234 — 0:0033 +0:0015 
But for the construction of final tables, from which the G- and ©-functions might be 
obtained for any value of ¢ by interpolation, the preceding Table III. is not sutticient. 
In order to facilitate the further computations, [ have searched for convenient formule 
of interpolation, so that the still troublesome direct calculations of the series may be 
avoided, and | have found that the following consideration leads to a convenient result. 
It may be easily shown that the differential equation 
PCT 2 me ae 
(l+w get e| 1- ert gn [v= 
is satisfied by the particular solutions 
a 
3 =Ses Fee ] 2\* es Be Paes 
(1+?) cos(V/e—4log(w+V/1+w2)) and = sin (,/c — Flog (w+ /1+w?)). 
renee! 
Now, the close similarity between this differential equation and the hyperbolic Seiche- 
equation, into which it converges if c is increased indefinitely, makes it probable that 
the solutions of the latter, 2.e. the @- and G-functions, are also closely related to the 
