HARMONIC ANALYSIS OF TIDAL OBSERVATIONS. 63 
XZ=(1— se") [p*qeos (x—2, + 37) +pq* cos (x +20,—}7) ] 
+ (1+ $e") pq (p?—9°) cos (x—37) 
+ $ep*g cos (x—3e,+a,4+47) 
t+ep*qv {$+} cos 2a} cos (y—7,+ Q—3r) 
+ 3epq (p?—q") cos (x+0,—2,—37) 
+ le? p3q cos (x—40,+2a,+47) 
+ io mep*g cos (x—20,—s+2h—p+ pr)’. . . . . (30) 
3 (NP + B22") =3 (p*—4p7@? +g) [1+ $e +38 cos (s,—a,) 
4?me cos (s—2h+p)+3m? cos (2s—2h) ] 
+2p?q° [(1— $e?) cos 20,+ fe cos (82,—a,)] (31) 
In these expressions 
sin 2a 
tan R=—~—, > :«tan Q=} tan a, 
qcot? £[—cos 2a, Fi 
The next step is to express the angles x, o,, a,, each of which increases 
uniformly with the time, in terms of the sidereal hour-angle or of the local 
mean time, and of the mean longitudes of the moon, and of the perigee. 
Fig. 2. 
P_Morbit _ 
Ectipiie 
Let M be the moon in the orbit. A the extremity of the A-axis fixed 
in the earth. 
g be the sidereal hour-angle. 
N the longitude of the node &. 
v the right ascension of the intersection I. 
£ the longitude ‘in the moon’s orbit’ of the intersection. 
t the inclination of the moon’s orbit to the ecliptic. 
w the obliquity of the ecliptic. 
s the moon’s mean longitude. 
p the mean longitude of the perigee.! 
Then (Fig. 2) g=Ar, v=1TI, <=TrR—gI, N=rQ. 
Now o,and a, haye been defined above as the moon’s mean longitude 
and the longitude of the: perigee, both measured in the orbit from the 
intersection I. 
* This p will easily be distinguished from the p used above to denote cos 31. 
