HARMONIC ANALYSIS OF TIDAL OBSERVATIONS. 85 
The following are the values of cos u, thus determined from (53 ') :— 
(a) cos 2(v—é), =1—?? ec) 
S10 @ 
(6) cos 2y,=1—7? —>+— 
=o 
(y) cos (28—»),=1—312(= FE 
9 
(6) cos (28+yv),=1 —ye(Eee 
sin w 
ea | 
eet eee 
E cos ro =1—127— 
(<) ‘ 4° sin? w 
(é) cos 25,=1—7? cot? 
The suffix , indicating the mean value. 
. 
Similarly the following are the J,’s or mean values of J :— 
aed 
» sin? 4w—cos w 
(a’) cos* 4J,=cos* tof 1 +30? om eee] 
cos? 50 
— 3 gin2 
(B') & (2’) sin? I,=sin? o [2 +72 ees] 
sin? w 
, : 1» (cos 2w 2 cos w 
(7) sin I, cos? $7,=sin w cos? $o [2 +i? ( - | 
_ 
sin? w cos? 4w 
© 
an2 
tae ; : : »f/cos 20 , 2 cos w 
(o’) sin I, sin? 3J,=sin sin? 5 [ u +4°( on) | 
9 
sin? w 
s1n 
(e’) sin I, cos J,=sin o cos w [1+3:? (cot? o—5)] 
On referring to schedules [B], it appears that (a) multiplied by (a’) is 
the mean value of the cos! $1 cos 2(v—£) which occurs in the semidiurnal 
terms; and so on with the other letters, two and two. 
Performing these 
multiplications, and putting 1—4i? in the results as equal to cos* 47, and 
1—3/? as equal to 1— sin? i, we find that the mean values are all unity 
for the following functions, viz. : 
cos* $T cos 2(v—£) sin? I cos 2 sin I cos? 
4I cos (2—v) 
cost 4w cost 47’ 
sin J sin? 37 cos (2§+,1) sin I cos I cos v 
5 7 SST eT 5 5 . 
sin? w (1—# sin? 7) sin w cos? dw cos* 50 
sin? I cos 25 
5 = 5 ; A _o TG ” 
sin w sin? So cost fi” sin w cos w (1—$ sin? #)’ sin® w cost {4 
Lastly, it is easy to show rigorously that the mean value of 
1— sin? I 
‘ (1—# sin? w) (1—$ sin? 7) 
is also unity. 
