234 
MR. S. S. HOUGH ON THE APPLICATION OF HARiMONIC 
type whose frequency approximates to the root of the equation L,, = 0 ; we have 
seen in the preceding sections how to evaluate + K;,+ 4 , . . . and H„_ 2 , . . . 
Also we have 
+ — (2r + 3) (2?’ + 5) K^4.2 j 
C,_2/C, = (2r - 3) (2r - 1) H,.., 
and therefore 
+ 0 = (2'n 3) {2n + 5) K„ 0,„ 
+ 4 = (2;; + 3) {2ii + 5) (2n + 7) {2n + 9) + o C„, 
0«_2 = {2ii — l){2n — 3) H ,,_2 C,,, 
a,_ 4 = (2n - l)(2n - 3) (2n - 5) (2n - 7) H,_ 2 H,_ 4 C,, 
Thus the height of the surface-waves is given by 
--+ ( 2 n - 1) (2n - 3) (2n - 5) {2?i - 7) H „_2 H „_4 P „_4 
+ {2n — 1) [2n — 3) H;i _2 P «_3 + P« -f- {2ti -1-3) (2n + 5) K/i +3 Pn+o 
_+ (2n -{- 3) {2n p 5) {2n -1-7) (2n -f- 9) K„+ 4 , P „+4 4- - • • 
where X is the root of the frequency-equation in question, and C„ is an arbitrary 
constant. 
Continuing with the particular numerical example dealt with in § 7, we take 
7 -, = 2-25394, 
or ^ = 1-5014 
2(0 
and deduce 
Lo 
-281944, 
L,2 = 
- -013199, 
L 4 -- 
-065179, 
11 
- -01G171, 
L, = 
-018021, 
11 
- -01814, 
Ls = 
Lio — ~ 
-000232, 
■ -008378. 
L,8 = 
- -01951, 
Neglecting Koq, we find 
log Kjg = 5 - 535 , log Kjg 
= n 5-7785, 
log K ^4 = n 4-0698, 
log Kio : 
= n 4-4397, 
log Kio 
= 9i 4-9812, 
and in like manner 
log Ho = 3-2UG4, log H 4 = 3-0458, log i p = 3-0753 ; 
