ON THE HARMONIC ANALYSIS OF TIDAL OBSERVATIONS. 49 
gain, 1 a dP) 
%: / J 3 3 
Ii+<all rt ea aes 
Hence, if (P’—1)/e denotes the value of (P—1)/e ata time a/(¢—7) 
earlier than that of observation, then 
Wan =1(p’—1). 
e 
Hence, in (23), 
1+ tan {(u—r)1'=1(P'—1) MrT 005 
where P’ is the ratio of the moon’s parallax to her mean parallax at a 
time 7,f—57°°3 tan 2(u—v)/(e—a). The period 57°°3 tan2(u—v)/(e—a) 
may be called ‘ the age of the parallactic inequality.’ 
In collecting results we shall write the sum 
M,+8,+K,+N+1L+R+T=A,. 
For reasons explained below we omit terms depending on the rate 
of change of solar parallax and declination. 
Then, from (15), (16), (18), (19), (23), (24), (25), (26), we have 
- cos2A Mcos 2(¥—p) + S cos 2(W,—Z) 
> cos?A, 
ee cos 20—Ky+ ones K/' cos 2(,—«) 
ao eerie —M tan?A, ) sin 2(b—p) 
- oe (P—1) cee ot oo 2b 8) 
an ey 62 ®) 05 2(v,-—Z) 
e, 
eosA 6L. aP _ Nsece2(u—v)+Lsec2(A—p)\ = 97) _ 
cos"A, o—a 7 (Sif e ) sin 2 #) 
(27) 
It may easily be shown, from Schedule B. i., 1883, that in the equi- 
librium theory K’—Mtan?A,=0, and 4M—(N+L)/e=0; hence the 
terms depending on rates of change of declination and parallax are small. 
This also shows that we were justified in neglecting the corresponding 
terms in the case of the sun. Also, since the faster tides are more 
augmented by kinetic action than the slow ones, the two functions, 
written above, which vanish in the equilibrium theory are normally 
actually positive. The formula (27) gives the complete expression for 
the semidiurnal tide in terms of hour-angles, declinations, and parallaxes, 
_ with the constants of the harmonic analysis. 
We shall now show that with rougher approximation (27) is reducible 
toa much simpler form. 
ce retardation of each tide should be approximately a constant, plus 
: E 
