= Oo - 
TOJASSIST WORKTON THE TIDES. 235 
It is now convenient to write these in the form 
A,=F cos (n—pT) 
B,=F sin (n—pT), } 
where 
F cos n=/'R ccs e' +f’’R cos e”’, 
F sin n=/f'R sin e'+/''R sin e”, 
and p is the equal to 24(¢—15n)°: it differs from the true speed in degrees per mean 
solar day only by a multiple of 360°. We shall speak of ¥, 7 and p as the ‘ reduced 
values’ of R,«ando. We now obtain 
(F/R)? =f"? +f"? +277" cos 15° ) 
and Yi ; { : . (2) 
tan (n—€')= sae ea te . 
f'+f'' cos 15°n 
As, however, we shall not have occasion to use values of N other than N=30 the 
relations between F and R, 7 and e may be given once for all in a numerical form 
for convenient values of p. The value of ¢—7 is practically the same for all values 
of n, but the ratio of R to F varies with n. 
n=0 n=l n=2 
P ream R/F R/F R/F 
0° 0° 1:0000 1:0000 1:0000 
1° 14-96° 1:0115 1:0101 1:0109 
ge 29-92° 10472 1:0444 1:0459 
3° 44-88° 1:1107 1:1063 1:1086 
4° 59°84° 1-2092 1°2027 12062 
5° 74:80° 1:3552 1:3461 1:3509 
6° 89-76° 15708 1°5582 15649 
Having obtained A,, B, for all the values of n, each pair of harmonic numbers 
is treated separately, so that we shall now omit the suffix 7. 
If p is zero then A and B are constants; if p is not zero then both A and B vary 
harmonically with the period 360/p days. We shall get what we may call ‘ conjugate 
constituents’ with the same period if the values of p are equal and opposite in sign. 
Thus R, and T, with p= + 0°9856 are conjugate with respect to S,. Now, in general, 
we shall have several constituents occurring together so that we shall have 
A =F, cos 1+ % {F, cos (n,—p,T) + F’, cos (n';+p;T)}, | 
ze 
Sy ws . (8) 
B = F, sin 4 +2 {F, sin (n,—p,T) + F’, sin (',+p,T)}, J 
s 
and we can consider p, as positive. These may be written as 
A = Aw +% (Az, cos p,T + As, sin p,T) 
ti : : . (4) 
B = Bs, + = (By, cos p,T + B,, sin p,T) 
= 
_ where 
Ag = Fy cos 7, By = F, sin 1, 
Ac, = F cos n,+ F’, cos 7’,, As, = F, sin n,—F’, sin 7’, { (6) 
Bey = —F, cos 9, + F’, cos 7’, Bey = F, sin n,+ F’, sin 7’, 
‘Therefore we have 
F, cos nx =3(A,,—B-y), F,, sin 7, = 4(A_, + Bs,) (6) 
F,’ cos n, = 3(Acr + Ber), F,’ sin n'y = a — Ag+ B,,) 
