During the last few years of his life Prof. Simon Newcomb was keenly interested in the problem of periodicities, and devised a new method for their investigation. This method is explained, and to some extent applied, in a paper entitled "A Search for Fluctuations in the Sun's Thermal Radiation through their Influence on Terrestrial Temperature." The importance of the question justifies a critical examination of the relationship of the older methods to that of Newcomb, and though I do not agree with his contention that his process gives us more than can be obtained from Fourier's analysis, it has the advantage of great simplicity in its numerical work, and should prove useful in a certain, though I am afraid, very limited field. Let
f
(
t
) represent a function of a variable which we may take to be the time, and let the average value of the function be zero. Newcomb examines the sum of the series
f
(
t
1
)
f
(
t
1
+ τ) +
f
(
t
2
)
f
(
t
2
+ τ) +
f
(
t
3
)
f
(
t
3
+ τ) + ..., where
t
1
,
t
2
, etc., are definite values of the variable which are taken to lie at equal distances from each other. If the function be periodic so as to repeat itself after an interval τ, the products are all squares and each term is positive. If, on the other hand, the periodic time be 2τ, each product will be negative and the sum itself therefore negative. It is easy to see that if τ be varied continuously the sum of the series passes through maxima and minima, and the maxima will indicated the periodic time, or any of its multiples.
III. Researches in spectrum-analysis in connexion with the spectrum of the Sun.—Part III
