Memoirs presented to the Cambridge philosophical society on the occasion of the jubilee of Sir George Gabriel Stokes, bart., Hon. LL. D., Hon. SC. D., Lucasian professor.

126 MR SCHUSTER, THE PERIODOGRAM OF MAGNETIC DECLINATION
so that the probability that the average of n values should lie between icE and
(ic + dK) E is
n^
- - n-1 e-K d/c.
(n- 1)!
If n is large, we may simplify the numerical calculation by putting approximately
according to Stirling's theorem
log (n - 1)! = (n - 1) log n - n + log 27r,
from which it follows that
nn )!=e /
(n -^r6 27r_
In order to illustrate the law according to which a gradually increasing number of
intensities tends to approach the value of the expectancy, I have plotted in Fig. 2
the curve
y= (n- 1)!n-1 e-n,
K
I2
A1
1
H A2
0 '5 i 1-5 2 2 5 3 3-5 4
Fig. 2.
for the three cases that n equals one, five, or fifty; fifty being the number on which
our Periodograph in the neighbourhood of the 26 day period rests.