10 
REV. S. HAUGHTON ON THE TIDES OF THE 
Multiplying together (26) and (32), we have, as a first approximation 
M ' cos m—s — i m — 
(0-41A— 0-18B) + (0-4lA+0-18B) cos2w£ 
(— 0-18A+0-41B) sin 2 nt 
M' cos (m — s — i m ) = + 0 • 3 3 M sin <f> 
+M (-0 •35 cos 0'29 sin <£) cos 2 nt 
-{-M ( — 0‘34 cos 0+0*30 sin </>) sin 2nt (34) 
Multiplying together (26) and (33) we find, also, 
M' sin (m— s— i m ) = 
(0-42A , -0-18B / ) + (0-42A , + 0-18B / ) cos 2 nt 
+ (-0-18A / +0'42B / ) sin 2 nt 
or, 
M' sin (pi—s—i m ) = — 0'4lM cos </> 
+M (—0*30 cos </)— 0’35 sin ty) cos 2 nt 
+ M (+0‘30 cos <f>— 0‘34 sin <f>) sin 2nt (35) 
Equations (34) and (35) are now to be converted into functions of u, which may be 
effected, in general, by means of the following expansions,' 5 " in which a is a proper 
fraction : — 
cos a x— 
sin a 7 t f 
• -(-2 a 
+ 
l 2 — a 2 
cos 2 x 
2 2 — a 2 
cos 3 x 
cos Ax 
4 2 — a 2 
_+&c. 
(36) 
sin a x= 
2 sin a tt 
+ 
l 2 — a 2 
2 sin 2 x 
' 2 2 — a 2 
3 sin 3 x 
3 2 — a 2 
4 sin 4 x 
4 2 — a 2 
,-(-&c. . 
( 37 ) 
I am indebted to Benjamin Williamson, E.T.C.D., for these formulae. 
