ON THE HARMONIC ANALYSIS OF TIDAL OBSERVATIONS. 45 
R.A. of the intersection of the equator with the lunar orbit, being 
denoted by y,. The initials of each tide are used to denote its height at 
any time. 
§ 3. Introduction of Howr-angles, Parallaxes, and Declinations. 
We must now get rid of the elements of the orbit and of the mean 
longitudes, and introduce hour-angles, declinations, and parallaxes. 
At the time ¢ let a, é, J be )’s R.A., and declination, and hour-angle. 
and a,, 6, w, ©’s R.A., and declination, and hour-angle. 
Let 7 be )’s longitude in her orbit measured from ‘ the intersection,’ 
and a—v, (v, being the » of 1883) be )’s R.A. measured from the 
intersection. _ 
The annexed figure exhibits the relation of the several angles to one 
another. 
Z M__ orB/T 
= T) fs 
a—7 EQUATOR 
The spherical triangle affords the relations 
tan («—yv,)=cosItanl, sne=sinIsind . . . . (1) 
From the. first of (1) we have, approximately, 
a=1+v,—tan*Z7sin2i” . . . . . . (2) 
Now, s—é is the moon’s mean longitude measured from I, and s—p is 
the mean anomaly. Hence, approximately, 
b==e— E+ Zemin(s—p) ys ee (8) 
And therefore, approximately, 
a=s+v,—F+2esin(s—p)—tan?3/sin2(s—é) .: . (4) 
Now, t+h being the sidereal hour-angle, 
i tla ey are télé. ‘verde (5) 
Therefore, from (4) and (5), 
t+h—s—(v,—£)=+2esin(s—p)—tan?5Tsin2(s~Z). . (6) 
By the second of (1) we have, approximately, 
cos?é = 1—}sin*?I+ $sin*Icos2(s—f) . . . . (7) 
Hence, if A be such a declination that cos?A is the mean value of cos? 6, 
we have 
cos?A = 1—tsin? I 
eee ei 
and cos?A,=1—4sin?o 
From this we have (neglecting terms in sin*4) the following relations: — 
cost4I=cos?A, sinIcos?4I=)/2sinAcosA, sin? [= 2sin? A, 
cos! w= cos?A,, sinwcos*zw= /2sinwcosw, sin?w=2sin? A, 
