408 PROFESSOR W. THOMSON ON THE REDUCTION OF 
Hence 
if uy 
s=ql FO at 
A, =3 fro cos aE 
and similarly we find ae F(¢) sin ga dt:— 
equations by which the coefficients in the double series of sines and cosines are 
expressed in terms of the values of the function supposed known from ¢=o0 to 
t=T. The amplitudes and epochs of the single harmonic terms of the chief — 
period and its submultiples are calculated from them, according to the following 
formula :— . 
Tan ¢.= B, 2 es +B)! 
ah ay a 
(or for logarithmic calculation, M,=A, sec ¢)), 
The preceding investigation is sufficient as a solution of the problem,—to 
find a complex harmonic function expressing a given arbitrary periodic function, 
when once we are assured that the problem is possible; and when we have this 
assurance, it proves that the resolution is determinate ; that is to say, that no other 
complex harmonic function than the one we have found can satisfy the conditions. 
For a thorough and most interesting analysis of the subject, supplying all that is — 
wanting to complete the investigation, and giving admirable views of the problem 
from all sides, the reader is referred to Fourter’s delightful treatise. A concise — 
and perfect synthetical investigation of the harmonic expression of an arbitrary 
periodic function is to be found in Poisson’s ‘‘ Theorie Mathématique de la 
Chaleur,” chap. vii. 
Il.—Periodie Variations of Terrestrial Temperature. 
7. If the whole surface of the earth were at each instant of uniform tempera- 
ture, and if this temperature were made to vary as a perfectly periodic function 
of the time, the temperature at any internal point must ultimately come to vary 
also as a periodic function of the time, with the same period, whatever may 
have been the initial distribution of temperature throughout the whole. Fourrer’s 
principles show how the periodic variation of internal temperature is to be con- 
ceived as following, with diminished amplitude and retarded phase, from the vary- 
ing temperature at the surface supposed given: and by his formule the precise — 
law according to which the amplitude would diminish and the phase would be — 
retarded, for points more and more remote from the surface, if the figure were — 
truly spherical and the substance homogeneous, is determined. 
8. The largest application of this theory to the earth as a whole is to the ana~— 

