422 
THE ASTRONOMER ROYAL ON THE DIURNAL AND ANNUAL 
Double three-hourly sums of Means of Luno-Diurnal Inequalities. 
Extent of Group 
in Lunar Hours. 
Western Declination. 
Horizontal Force. 
Years of large 
solar curves. 
Years of small 
solar curves. 
Years of large 
solar curves. 
Years of small 
solar curves. 
h h 
0 to 2 
+ 1-33 
+ 6-90 
+ 0-000284 
+ 0-000182 
3 to 5 
+ 0-10 
—0-34 
+ 
176 
+ 
91 
6 to 8 
-1*31 
— 0-86 
— 
313 
— 
372 
9 to 11 
-0-17 
+ 0-72 
— 
86 
— 
19 
12 to 14 
+ 1-35 
+ 0-94 
+ 
387 
+ 
354 
15 to 17 
+ 0*27 
-0*33 
+ 
147 
+ 
156 
18 to 20 
-1*14 
-0-89 
— 
387 
— 
201 
21 to 23 
-0-43 
-0-11 
— 
215 
- 
169 
The inequalities are so evidently semidiurnal* that we may at once proceed to treat 
them on that assumption. Then each of the groups corresponds to a quadrant of luno- 
semidiurnal tide, and the first quadrant must be understood to begin at23^ h lunar time, 
or, in arc, 15° before the moon is on the meridian. And when, from the numbers above, 
we shall have ascertained the argument of the luno-semidiurnal tide as measured from 
that lunar epoch 23^ h , we must represent the argument as measured from lunar noon, or 
the argument in lunar time, by adding 15° to the argument reckoned from 23| h . 
In each of the semidiurnal tides, the coefficient of the sine of double lunar angle from 
23^ h may be found by taking 1 st number -j-2 d number —3 d number —4 th number, and 
dividing by 8 X {sin 15°-f-sin 45° -f- sin 75°} = 15‘4544. And the coefficient of the cosine 
may be found by summing l 8t number -}-4 th number — 2 d number —3 d number, and using 
the same divisor. The sum of a multiple of sine and a multiple of cosine may then be 
converted into a single sine; and the correction +15° may be applied, to render the 
argument measurable from lunar noon. Thus we obtain, as expressions for the Lunar 
Inequality, 
In Western Declination from North, 
Years of large 
solar curves 
Years of small 
solar curves 
( First Semidiurnal Wave, 0'-246 x sine (double moon’s hour-angle +39° + 15°) 
Second Semidiurnal Wave, 0'-237 xsine (double moon’s hour-angle +31° + 15°) 
Mean, 0' - 242 x sine (double moon’s hour-angle + 50°) 
{ First Semidiurnal Wave, 0'T88 x sine (double moon’s hour-angle + 14°+ 15°) 
Second Semidiurnal Wave, 0'Y69 xsine (double moon’s hour-angle + 38° + 15°) 
Mean, 0'-179 x sine (double moon’s hour-angle + 41°) 
* The numbers in each of the four columns of figures above may be resolved into a semidiurnal and a diurnal 
series, as follows : — 
+ 1-34 
-0-01 
+ 0-92 
-0-02 
+ 335 
-51 
+268 
-86 
+0-18 
-0-08 
-0-34 
0-00 
+ 161 
+ 15 
+ 124 
-32 
-1-22 
-0-08 
-0-88 
+ 0-02 
—350 
+37 
-286 
-86 
-0-30 
+0-13 
+0-30 
+ 0-42 
-150 
+ 65 
- 94 
+ 75 
+ 1-34 
+ 0-01 
+0-92 
+ 0-02 
+ 335 
+ 51 
+ 268 
+ 86 
+0-18 
+ 0-08 
-0-34 
0-00 
+ 161 
-15 
+ 124 
+32 
-1-22 
+0-08 
-0-88 
-0-02 
-350 
-37 
-286 
+ 86 
-0-30 
-0-13 
+0-30 
—0-42 
-150 
— 65 
- 94 
-75 
