PROFESSOR A. SCHUSTER ON THE PERIODICITIES OF SUNSPOTS. 73 
coincidence at the midway point as regards time, i.e., in the present case at a time 
T + jfTl T. 
We begin always with the same epoch, but provide for the possibility of sub¬ 
dividing the time interval so as to obtain separately the periodogram of different 
parts. We therefore must not neglect r, but we may put it equal to an integral 
multiple of the trial period, or say to mT. If NT is the total time interval considered, 
we have thus n + m = N, so that the time at which coincidence of phase is required is 
^-(N + m)T. If now cos (gt—e) is the true periodicity and cos (rd — c/>) that of the 
trial period which is to be adjusted so as to fit the true period most nearly, we 
must put 
\g (N + m) T-c = (N + m) T -<f >, 
therefore 
e -<£ = h (9~ K ) (N+m) T = *5^(N + m). 
This determines the difference in phase between that belonging to the true period 
and that determined with a slightly erroneous trial period. Table III. gives, e.g., 
the angle </> in the neighbourhood of the 11-year period, and shows how the phase 
of the true periodicity may be determined. This periodicity, it will appear, is only 
prominent in the second portion of the complete time interval considered. The first 
column gh es the times of the trial periods in years. The second and third columns 
give the values of N and m ; in the fourth column is given the phase as calculated 
by the trial period, the fifth gives (e — </>) the difference of the phases between the 
trial and true periods calculated by means of the above equation on the supposition 
that the true period was 11T25 years. The last column gives the calculated phase 
( >f the true period. In calculating the last column from the two preceding ones 
it must be borne in mind that 360° or any multiple thereof may be added or 
subtracted from an angle representing a phase without alteration of its significant 
value. 
Table III. 
T. 
9-75 
10-00 
10-25 
10-50 
10-75 
11 
11 
11 
11 
00 
25 
50 
75 
12-00 
12-25 
12-50 
12-75 
N. 
15 
15 
14 
14 
14 
14 
13 
13 
13 
12 
12 
12 
11 
m. 
</>. 
€ - 
€. 
7 
107° 
- 129° 
-22° 
7 
35 
41 
- 6 
7 
306 
297 
+ 9 
7 
221 
212 
9 
7 
142 
127 
15 
7 
50 
- 42 
8 
0 
335 
+ 38 
13 
6 
250 
115 
5 
6 
180 
192 
12 
6 
107 
255 
2 
6 
37 
328 
+ 5 
6 
317 
401 
_ 2 
5 
259 
+ 421 
-40 
VOL. OCVI. — A. 
