ON THE HARMONIC ANALYSIS OF TIDAL OBSERVATIONS. 51 
For the solar parallactic term, 
de,S cos (h--p,) cos 2[b,—¢]. 
Then omitting the terms depending on changes of declination and 
parallax, we have as an approximation, 
hy=fM | cos 2(W—p) +2 tan? $7 cos 2[s—cer—E] cos 2(W—Z) 
+3e cos [s—p—w(s—az)]| cos 2[V—p+ 2e(0— =i] 
+S | 1+2 tan? jw cos 2h+Be,cos (h—p,) | eos 2-2). . (28) 
| = 
In the equilibrium theory we have the lunar semidiurnal tide depend- 
ing on 7~* cos? 6 cos 2. Now it is obvious that cos?d introduces a 
factor 142 tan? I cos 2 (s—é), and 7? a factor 1+3e cos (s—p). Thus, 
if we could have foreseen the exact disturbance introduced by friction and 
other causes in the various angles, the formula (28) might have been 
established at once ; but it seems to have been necessary to have recourse 
to the complete development in order to find how the age of the tide will 
enter. 
§ 4. Reference to Time of Moon's Transit. 
It has been usual to refer the tide to the time of moon’s transit, and 
we shall now proceed to the transformations necessary to do so. 
cos” A /cos? A , goes through its oscillation about the value unity in 
19 years ; it is therefore convenient to write for, say, a whole year, 
2 
u,= 4 uw ) 
cos" A, 
9 
oe 2A | 
and similarly, N,=°* & ; 
Vagtr cos?A,;) fF 2 2 +» (29) 
__cos?A 
°eos?A 77 J 
/ 
We also observe that K” and K/’, being the lunar and solar parts of 
the mean K, tide, and their ratio being ‘464 (Report, 1883), 
K" =68303 K,, K/'='31697 K, . . . . (30) 
It will also be seen that in all the terms arising from the sun, excepting 
that in K,’, the argument of the cosine is 2(~,—2). It will be con- 
venient, and sufficiently accurate for all practical purposes, to replace 
«by ¢ in this solar declinational term K/’. 
We shall now proceed to refer the tide to the moon’s transit at the 
place of observation. 
Let a,, h, be )’s R.A and ©’s mean longitude at )’s transit—say 
upper transit, for distinctness. Then the local time of transit is given by 
the vanishing of W, and since ~=t+h—a, it follows that the time-angle 
of )’s transit (at 15° to the hour) is a,—h,. 
| Now let r (mean solar hours) be the interval after transit to which the 
_time-angle ¢ refers; then, since 
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