On the Reduction of Observations of Underground Temperature. 23 



mical speculation and research, remembering that my own pub- 

 lications on the subject, which cover the whole ground, were 

 some years earlier than those of Williamson, Gerhardt, Wurtz, 

 or Kolbe. 



Montreal, January 1861. 



IV. On the Reduction of Observations of Underground Tempera- 

 ture ; with Application to Professor Forbes's Edinburgh Obser- 

 vations, and the continued Calton Hill Series. By Professor 

 William Thomson, F.R.S.* 



I. Analysis of Periodic Variations. 

 1. ii^VERY purely periodical function is, as is well known, 



JL-J expressible by means of a series of constant coefficients 

 multiplying sines and cosines of the independent 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 Bernoulli, partially given by La Grange, and 

 perfected by Fourier. 



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 vari- 

 able. The harmonic curve is the well-known name applied to 

 the graphic representation, on the ordinary Cartesian system, of 

 what I am 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 variable multiplied by a constant factor, with a con- 

 stant 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 



* From the Transactions of the Royal Society of Edinburgh, vol. xxii. 

 part 2. Communicated by the Author. 



