Observations of Underground Temperature. 25 



tion of which the amplitude and epoeh are derived from the ampli- 

 tude and epochs of the given functions, in the same manner as 

 the magnitude and inclination to a fixed line of reference, of the 

 resultant of any number of forces in one plane, are derived from 

 the magnitudes and the inclinations to the same line of reference 

 of the given forces. 



Cor. 3. The physical principle of the superposition of sounds 

 being admitted, any number of simple unisons of one period co- 

 existing, produce one simple unison of the same period, of which 

 the intensity (measured by the square of the amplitude) and the 

 epoch are determined in the manner just specified. 



Cor. 4. The sum of any number of simple harmonic functions 

 of one period vanishes for every argument, if it vanishes for any 

 two arguments not differing by a semicircumference, or by some 

 multiple of a semicircumference. 



Cor. 5. The co-existence of perfect unisons may constitute 

 perfect silence. 



Cor. 6. A simple harmonic function of any epoch may be 

 resolved into the sum of two whose epochs are respectively zero 

 and a quarter period, and whose amplitudes are respectively 

 equal to the value of the given function for the arguments zero 

 and a quarter period respectively. 



4. Complex Harmonic Functions. — Harmonic functions of dif- 

 ferent periods added can never produce a simple harmonic func- 

 tion. If their periods are commensurable, their sum may be 

 called a complex harmonic function. 



Cor. A complex harmonic function is the proper expression 

 for a perfect harmony in music. 



5. Expressibility of Arbitrary Functions by Trigonometrical 

 series. 



Prop. A complex harmonic function, with a constant term 

 added, is the proper expression, in mathematical language, for 

 any arbitrary periodic function. 



6. Investigation of the Trigonometrical Series expressing an 

 Arbitrary Function. — Any arbitrary periodic function whatever 

 being given, the amplitudes and epochs of the terms of a com- 

 plex harmonic function, which shall be equal to it for every 

 value of the independent variable, may be investigated by the 

 " method of indeterminate coefficients/'' applied to determine an 

 infinite number of coefficients from an infinite number of equa- 

 tions of condition, by the assistance of the integral calculus as 

 follows : — 



Let ~F(t) denote the function, and T its period. We must 

 suppose the value of ~F(t) known for every value of t, from t=o 

 to t=T. Let M denote the constant term, and let M 3 , M 2 , Mg, 

 &c. denote the amplitudes, and e l} e 2 , eg, &c. the epochs of the 



