HARMONIC ANALYSIS OF TIDAL OBSERVATIONS. 61 
In this case also the values of the coefficients are actually considerably 
greater than the amounts as computed from the first terms; and regard 
must be paid to this, as in the case of the Evection, when the values of 
the coefficients in the tidal expressions are computed. According to Pro- 
fessor Adams, the full values of the coefficients are, in longitude -011489, 
and in c/r °008249. 
We have now obtained in (20), (26), (27), the complete expressions 
for the X-Y-Z functions in the shape of aseries of simple time-harmonics ; 
but they are not yet in a form in which the ordinary astronomical formule 
are applicable. 
Further substitutions will now be made, and we shall pass from the 
potential to the height of tide generated by the forces corresponding to 
that potential. 
The axes fixed in the earth may be taken to have their extremities as 
follows : 
The axis A on the equator in the meridian of the place of observation 
of the tides ; the axis B in the equator 90° east of A; the axis C at the 
north pole. 
Now £, n, £ are the direction-cosines of the place of observation, and 
if X be the latitude of that place, we have 
E=cos Ae) =O} ef==sin A, 
Thus 
E—n*=cos*\, én=0, ni=0, 2éfZ=—sin 2\, 3 (€?+n?—22*)=1-—sin2A. 
3 2 
Then writing a for the earth’s radius, the expression (10) for V at 
the place of observation becomes 
2 
eraera [3 cos?\ (xX?*— Y?)+sin 2.XZ 
—e¢ 
+2 (4—sin’) 2 (X24 Y?—272)} 
The X-Y-Z functions being simple time-harmonics, the principle of 
forced vibrations allows us to conclude that the forces corresponding to V 
will generate oscillations in the ocean of the same periods and types as 
the terms in V, but of unknown amplitudes and phases. 
Now let 4°—2?, NZ, 3(X°?+°—2Z?) be three functions, having | 
respectively similar forms to those of 
X?— y?2 XZ aed (50 Y' —22") 
(1—e?)” (1—e?)3 3 (1—e?)3 ? 
but differing from them in that the argument of each of the simple time- 
harmonics has some angle subtracted from it, and that the term is 
multiplied by a numerical factor. 
Then if g be gravity, and h the height of tide at the place of observa- 
tion we must have 
bate [5 cos?\ (X?—337)+sin ANZ 
+$ (}—sin®d) 4 (N24 92-22%)] (28) 
2 / 3 
The factor er may be more conveniently written 3} x ( =) a, where 
c 
