222 
MR. S. S. HOUGH OH THE APPLICATION OF HARMONIC 
The law of variable depth which we have assumed appears to be the most general 
law which will lead to a difference-relation connecting the successive C’s of order not 
higher than the second. We shall confine ourselves chiefly to the case where the 
depth is uniform, but the following remark with respect to the more general case 
seems worthy of attention. 
If we put for brevity 
^ _ 1 — n{n + 1)_ 1 — n{n + 1) 
~ {-In -H1) {2n -f .s) ’ '^"“2 ~ ~ 72 «"^ny( 2 wny~ ’ 
_ /2 — 1 2 {1 — n{n + 1) /^„/4&)-ft-} ^ leg,, 
~ n {n +[1) {2n - 1) {2n + 8) 4(»k/-^ ’ 
and suppose all the y’s zero, so that there is no disturbing force, equations ( 23 a) 
mav be written 
,/ 
— L^O] T^^Cg = 0, 
— L3C3 T^gCg = 0, 
4C3 - L5C5 4 - ^507 = 0, 
Now, suppose I in the expression Z: + Z (1 —/x®) for the depth is of the form 
, where r is an integer, and for greater definiteness let us suppose that r 
= 0 and y],_o = 0 , and therefore the equations ( 24 ) will all be satisfied if 
~ L 2 C 3 = 0 , 
— L j,G 4 -b = Oj 
£._ 4 CV -4 — L.-2C,-2 = 0, 
|^,._2C,._2 — L,.C,. + ') 7 rC,. 4.2 = 0, 
“ '^r+2^y+2 "b Vr+2^r+i — b, 
l’i- + 2Gi. + 2 L,.^.4C,. + 4 “b '>^}- + 4 G,. + 6 h, 
4 &)kr 
r(r + l)g, 
is even. 
Then ir 
~ L2C2 + >7304 
~ LgCc -b 
= 00 
= 0 , 
= 0 , 
. ( 24 ). 
J 
end all the C’s with odd suffixes vanish. 
Further, these will be satisfied if C,.+2> Cr+4, . . . are all zero, provided X is a root of 
the equation 
