
OBSERVATIONS OF UNDERGROUND TEMPERATURE. 407 
4. Complex Harmonic Functions.—Harmonic functions of different periods 
added, can never produce a simple harmonic function. If their periods are com- 
mensurable their sum may be called a complex harmonic function. 
Cor. A complex harmonic function is the proper expression for a parted 
harmony in music. 
5. Hxpressibility 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 Func- 
tion.—Any arbitrary periodic function whatever being given, the amplitudes and 
epochs of the terms of a complex 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 equations of condition, by the assist- 
ance of the integral calculus, as follows :— 
Let F(¢) denote the function, and T its period. We must suppose the value 
of F(z) known for every value of ¢, from ¢=o tot=T. Let M, denote the constant 
term, and let M,, M,, M,, &c., denote the amplitudes, and ¢,, ¢,, ¢,, &c., the 
epochs of the successive terms of the complex harmonic functions by which it is 
to be expressed ; that is to say, let these constants be such that 
9 
F(¢)=M, +4 M, cos Cr-s) + M, cos (“r —*) + M, cos (or -s) + &. 
Then, expanding each cosine by the ordinary formula, and assuming 
M, coss,=A,, M, coss,=A,, &e. 
M, sins, =B,, M, sins, =B,, &e. 
we have 
Qart Agr ist 
F@)=A,+ A, cos ae + Ay cos ar tA cos + tee. 
+B, sin =e + B, sin 7 +B, sin 7 + be. 
Multiplying 

tegrating from ¢=o0 to t=T, we ise 
T a AN ig 
[x cos ear cag Wad ; 
iS T 
=A, x4T, when 7 is any integer ; 

=A,xT, when 7 =0. 
