406 PROFESSOR W. THOMSON ON THE REDUCTION OF 
the period is equal to = Thus, if T denote the period, < the epoch in angular 
measure, and ¢ the independent variable, the argument proper for a cosine is 
Qat _, 
T ? 
and the argument for a sine 
Qat T 
cere 2) 
3. Composition and Resolution of Simple Harmonic Functions of one Period. 
Prop. The sum of any two simple harmonic functions of one period is equal 
to one simple harmonic function whose amplitude is the diagonal of a parallelo- 
gram described upon lines drawn from one point to lengths equal to the ampli- 
tudes of the given functions, at angles measured from a fixed line of reference 
equal to their epochs, and whose epoch is the inclination of the same diagonal to 
the same line of reference. 
Cor. 1. If A, A’ be the amplitudes of two simple harmonic functions of equal 
period, and ¢, «’ their epochs; that is to say, if A cos (mt—e), A’ cos (mt—¢) be 
two simple harmonic functions; the one simple harmonic function equal to their 
sum has for its amplitude and its epoch the following values respectively :— 
amplitude) {(A cose+A’ cos ’)?+(A sine+A’sin ¢)?}4; or {A?+2AA’ cos (¢—«) + A?}5 
Pp 
—1 A sin s+ A’ sin ¢' 
(epoch) A cos «+ A’ cos ¢ 
Cor. 2. Any number of simple harmonic functions, of equal period, added 
together, are equivalent to a single harmonic function of which amplitude and 
epoch are derived from the amplitude 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 magni- 
tudes and the inclinations to the same line of reference of the given forces. 
Cor. 8. 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 semi-circumference, or by some multiple of a semi-circumference. 
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. ’ 

