104 
CHRISTIE. 
Summing, dividing by v, and putting 
UH 0 +H 1 + . . . +H v _ i )=T, 
1 . sin v ft 
2 ^ T = /3,, 7i^ r = W '’ 
uc= C, us = S, ua = A, 
we obtain 
T— h + 2 (O r cos i t t + S r sin i T t) 
r 
= h 4- 2* ^4 r cos ( i r t — e r ) 
( 2 ) 
h 
For every value of t employed in the numerical summa¬ 
tions the terms in (2) arising from tides other than the one 
sought may be computed and removed whenever their de¬ 
fining elements are known, but the process would be very 
laborious, and for the present purpose is unnecessary. 
Let m be a positive integer, j the speed and mvr the com¬ 
plete period assumed for the tide sought, and put 
2 (G r cos i r t + Sr sin i r t) — 2 0 r , 
r r 
where i r no longer includes j ; then (2) becomes 
h + Cj cosj t + S i sinj t G 2j cos 2 jt + . . . — (T 7 — 2 0 t ) — 0. 
m — 1 m — 3 
Putting in this successively^ 
m — 1 
m 
m 
• + 
G 
m 
t r, where n — 180°, we have 
S 
"J _ nyt "I _ivvj 1 _ ryv) 
1 + cos -7i -f sin - 7t -f cos 2 ——— tt + ... — ( T — 20 r ) 1 _ m = 0 
2 2 
m 
m 
m 
1 -j- cos 
m 
, . 3 — m , 0 3 — m , V/1N 
tt + sin ——- X +cos 2 ——— * + . . . — (T— 20 r ) 3 _ 
m 
m 
1 + cos 
m- 
m 
1 . m 
— 7r -j- sm — 
m 
1 . 0 m 
— tt -f cos 2 — 
m 
«+ . ..-{T-20X-x= 0; 
