ON LINEAR DIFFERENTIAL EQUATIONS OF THE SECOND ORDER. 661 
UNINODAL SEICHE. 
nm=2,n(n—1)=1.2 
2 
5 ns (1 ~ wu?) \ sin mt, £= 7 sin my 
of 2 = (1 — w?) \ sin ft ee ad, 
BINoDAL SEICHE. 
n=3,n(n—1)=2.3 
QA a3 { . (ewe 
Dad eee (aay eae t 
a Bh dui) (1 — w?) psinn, € 7 tw Sit Ma 
2 
Dees eB =)? } sin not, € = *(3u? = 1) sin not - 
a dw? | 2 a 2 
TRINODAL SEICHE. 
n=4, n(n—1)=4.3 
A 4 { 3 : A z 
2.4.6 €= — 6 dwt | (1 — w?) } sin mat; €= — ah — 52) sin not (25) 
2A da 
2.4.6 f= - 
: A : 
e aa (1 - w?)" f sin mt; €= ~ gq (12 — 200? )o sin not . 
QUADRINODAL SEICHE. 
n=5, n(u—1)=5.4 
5 
2.4.6.8 _— 4 (1 —w?)* } Sin M4 ; g= 5 (Tu? — 8) sin 24t 
2 7A © § awe b singe; c= (3508 - 300 +3) sin n,f 
2.4.6.8 =. amas — w*) sin ",t ; ba oak dwt — 30w? + 3) sin n,t. 
The positions of the nodes are found from the equation ¢=0, and _ hence 
| (ql - wy" t =0 for n-nodal seiche, and since this differential-quotient is proportional 
to the LecenpRE-polynomial P,,(w) , the nodes are also determined by the equation 
P,(w)=0, 
where P,,(w) may be defined as the coetficient of h” in the expansion 
ore > P,(w)h”. 
n=0 
Turning now to the hyperbolic Seiche-functions, we obtain convenient expressions 
im the form of hypergeometric series by substituting in equation (3) wi for wand ni—a 
for n, 7 being the imaginary unit root ./—1. We have then 
ay dy 5 
(1 +02) 54 + (20+ lw 5 +(n?+a*)y=0, (26) 
the particular solutions of which may at once be taken from (17). Most of these series 
