TRANSACTIONS OF SECTION A, 417 
Pearson has investigated the now well-known method of correlation to detect hidden 
connections between sets of variables. It is the object of this note to show that these 
methods are connected, and to suggest a slight modification in Professor Schuster’s 
measure of the intensity of the periodic movement. 
2. Suppose we have a series of m numbers and are looking for a periodic repetition 
in groups of p. Then gua periodic the numbers will be comparable with the periodic 
series obtained by giving all values from 1 to » to k in a cos ka + 6 sin ka, where 
a=2rn/p, This suggests that we may obtain the values of a and b by correlating 
the given series 21, X) . . . 2», with the two series S cos ka and § sin ka. 
For this purpose we require the standard deviations of the three series. That of 
the given series may be found in the usual way : let it be s, 
u 
Since = a ka=0, and for the present we shall assume that n is a multiple 
k=1 
of p, the mean value of sine ka is zero, and the squared standard deviations are 
AS SR ee ees Ore =y=1/V72. 
Na Ny 
p 
3. Since the sum of the products = cos ka sin ka=0, there is no correlation between 
1 
the two trigonometrical series. 
We now require the mean products of the deviations of the given and trigonome- 
trical series, term by term, for the evaluation of the coefficient of correlation. These 
n 
= Ln mee Se ‘ z 
are | 2 (x; —x) cos ka and a = (x; —x) sin ka, where xis the mean of the given series, 
1 1 
and a;,—2 is the deviation. Since, however, >a cos ka=2% cos ka = 0 = Szsin ka, 
we may operate on the original series in place of the deviations of its terms, 
n n 
Professor Schuster writes A, = >x,cos ka, B, = 2a, sin ka and usesI = A? + B; 
1 1 ‘ 
as a measure of the intensity of the periodic movement. In Pearson’s method, the 
coefficients of correlation are r= A, /nys = A, /2/ns and r = B, V2/ns, and the 
coefficient of total correlation is R, where 
R? = r? +r? = 2(A5 + Bi)/n? s? = 21/n? 8°, 
R or R? is proposed as a modified measure of the precision of the periodicity. It 
differs from Schuster’s criterion in having n? s°/2 as a divisor of his I, and from the 
criterion adopted by Turner (op. cit.) in replacing his 3x by n3(~—z)?. It has the 
advantage of being more absolute than either of the other two. Personally I prefer 
R* to R, as giving a less exaggerated idea of the closeness of fit. 
The equation of regression is 
= 9 
&—x = (rs/y) cos ka -+- (r’ sc) sin ka = ne cos ka + = B, sin ka, 
the well-known result of harmonic analysis. 
4. Applying these results to the first and fifth groups given by Professor Turner 
last year, we find 
First Group. Fifth Group, 
: 4:77 3 
Mean » é F 5 : i f : 27-63 
Standard deviation ; 5 : : : , 2°76 8°53 
r : ‘ 6 : r 3 F , ; . —0°205 +0°213 
r ~~ PS A ‘ : , ‘ - p - —0°279 +0°128 
Re os : z : 3 : : 3 . - 0°34. 0:25 
Probable error of R : A 3 F : - +0:10 +0-11 
Compared with its probable error, R is not so large in either case as to prove 
periodicity beyond a doubt. 
5. So far only integral periods have been considered. When non-integral periods 
are suspected they may be approximately determined by considering consecutive 
values of R and interpolating, by a quadratic formula, to obtain that value of the 
period for which R isa maximum. This process would fail if there were two periods 
in the interval between two consecutive integral periods. Then it would be necessary 
to calculate R for periods of the form p, p + 4, p + 1, or PP+ P+% p+, &e., as 
shown by Professor Turner (op. cit. p. 335). 
912. EL 
