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XVII.—On the Reduction of Observations of Underground Temperature ; with 
Application to Professor Forbes’ Edinburgh Observations, and the continued 
Calton Hill Series. By Professor WitLIAM THomson. 
(Read 30th April 1860.) 
I.—Analysis of Periodic Variations. 
1. Every purely periodical function is, as is well known, expressible by means 
of a series of constant coefficients multiplying sines and cosines of the indepen- 
dent variable with a constant factor and its multiples. This important truth was 
arrived at by an admirable piece of mathematical analysis, called for by DANIEL 
BERNOUILLI, partially given by La GRANGE, and perfected by FourtEr. 
2. To simplify my references to the mathematical propositions of this theory, 
I shall commence by laying down the following definitions :— 
Def. 1. A simple harmonic function is a function which varies as the sine or 
cosine of the independent variable, or of an angle varying in simple proportion with 
the independent variable. The harmonic curve is the well known name applied 
to the graphic representation, on the ordinary Cartesian system, of what Iam now 
defining as a simple harmonic function. It is the form of a string vibrating in 
such a manner as to give the simplest and smoothest possible character of 
sound ; and, in this case, the displacement of each particle of the string is a har- 
monic function of the time, besides being a harmonic function of the distance of 
its position of equilibrium from either end of the string. The sound in this case 
may be called a perfect unison. 
Def. 2. The argument of a simple harmonic function is the angle to the sine or 
cosine of which it is proportional. 
Cor. The argument of a harmonic function is equal to the independent vari- 
able multiplied by a constant factor, with a constant added ; that is to say, it may 
be any linear function of the independent variable. 
Def. 3. When time is the independent variable, the epoch is the interval 
which elapses from the era of reckoning till the function first acquires a maxi- 
mum value. The augmentation of argument corresponding to that interval will 
be called “the epoch in angular measure,” or simply “ the epoch” when no am- 
biguity can exist as to what is meant. 
Def. 4. The period of a simple harmonic function is the augmentation which 
the independent variable must receive to increase the argument by a circum- 
ference. 
Cor. If c denote the coefficient of the independent variable in the argument, 
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