8 
REV. S. HAUGHTON ON THE TIDES OF THE 
/ \ 5 
:=nt-\-[ 2e— ^ j sin ( nt )-\ sin (2w£) 
13 
+j2 e3 sin (3wtf)+&c. 
( 25 ) 
Substituting this value in (24) and using the common expansions, 
cos *= 1 -^ 2 + f^ 3 r 4 -&c. 
£C 3 
sin x=x— Y7^g+&c. 
we find 
sin 2p,= — 0’04 — 0’35 sin (nt) — 0‘02 sin (2 nt) 
-f-0'82 cos ^+0'04 cos 2nt-\-&c (25) 
Multiplying together (22) and (25) we find 
M'=M (~^j sin 2 /a 
=M ( — 0 , 06+0 , 82 cos nt+0'02 cos 2 nt 
— 0'35 sin nt— O'Ol sin 2nt-\-kc (26) 
3°. We have now to find values in terms of nt for cos and sin 
This may be done as follows : — 
m— -\-kt — m! 
S— -\-Jct — s' 
where kt is the angle due to the rotation of the earth, and m' , s', are the proper motions 
in right ascension of the moon and sun ; therefore 
m — s=s' — m' 
We may find m! in terms of v, as follows : — If n be the angle between perigee and 
ascending node, and c be the angle between conjunction and ascending node, we have 
tan (m — c)=cos I tan (v — n) 
1 
cos (m' — c)= -v/l-f- cos 2 1 tan 2 (v—n) 
. . , , cos I tan (v— n) 
sm (m —c) = 2 T -— 2 , 
yi+ cos 2 1 tan 2 ( v — n) 
cos (m' — c) = 
sin ( m ' — c) = 
cos ( v — «) 
s/1— sin 2 1 sin 2 (v — n) 
cos I sin (v—n) 
y/l — sin 2 1 sin 2 (v — n) 
( 27 ) 
