ON LINEAR DIFFERENTIAL EQUATIONS OF THE SECOND ORDER. 663 
_ From equation (26) we see that the Seiche-equation, for which a= —#, lies between 
the two equations : 
a? dl 
(L410?) 4 — wh + (2+ ly =0 e.=n +1) 
Be ee 2 a 
(l+w Vag t aay t Py = 0 ~ Opa 
he particular solutions of these corresponding to the hyperbolic Seiche Cosine and 
che Sine are respectively : 
€_(w) = - sin [n log (w+ /1+w?)] + /1+w? cos [n log (w+ J/1+ a) 
6,(w) 
(Jew 
n Sy(w) 
cos [n log (w+ /1+w?)] 
(29) 
~cos [n log (w+ /1+w?)] + J1+w? sin [n log (w+ /1+w?)] 
ll 
sin [n log (w+ /1+w?)] 
We may also at once write down the corresponding particular integrals of the two 
ther differential equations (a = 1 and 2): 
d? d 
(1+ 02) 54 + 805" + (n? + 1)y =0 c= 1? +1 
(1 +u2/4 TY + bw ne i 0, cC,=n?+4 (30) 
G,(w) = (1 + w?) cos [m log (w+ J1 +w*)] 
Cy(u) = (14 0%)-# | V1 +0? cos [n log (w+ J1+@)] -= sin [n log (w + i+w } 
(31) 
— m S(w) = (1+ w*) sin [nlog (w+ J/1+v*)] 
(n+ +5) S,(u) = = +0) J/1 +0? sin [n log (w+ J/1+0)] + — cos [n log (e+ Jiu) } 
inthe other hand, the solutions (29) and (31) are expressed by the following 
eometric series : 
(S-FUNCTIONS. 
1) a Da eS a wee? 2.4) 96 
1.2 Kosa 1.2.3.4.5.6 
Gs see 2) Nes Co(Cy + 2.2) (Cy + 4.4) Keak: 
Tenino oS esieoiadl 1.2.3.4.5.6 i 
pede: supe GNGTe sae 4 _ (+ 4.2)(c, + 6.4), ieee? 
1.2.3.4 1.2.3.4.5.6 
hes £9 yp 4 Colla + 8.2 2), wes Co(Cy + 6.2) (Cy + 8.4) 
Mtb 6 ga 
T2 1.2.3.4 (Oe 
