24 Prof. W. Thomson on the Reduction of 



the interval which elapses from the era of reckoning till the 

 function first acquires a maximum value. The augmentation of 

 argument corresponding to that interval will be called "the 

 epoch in angular measure," or simply "the epoch" when no 

 ambiguity 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 circumference. 



Cor. If c denote the coefficient of the independent variable in 



o 

 the argument, the period is equal to — . Thus if T denote the 



period, e the epoch in angular measure, and t the independent 

 variable, the argument proper for a cosine is 



2irt 



and the argument for a sine, 



2irt 7T 



T 6+ 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 parallelogram described upon lines drawn 

 from one point to lengths equal to the amplitudes 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 e, e' their epochs, that is to say, 

 if A cos (mt—e), A' cos (mt — e 1 ) be two simple harmonic func- 

 tions, the one simple harmonic function equal to their sum has 

 for its amplitude and its epoch the following values respect- 

 ively :— 



(amplitude) { (A cos e + A' cos e') 2 + (A sin c + A' sin e') 2 } 1, 



or {A 2 + 2AA'cos(e'-e)+A' 2 }ij 



, A sin e + A' sin e' 



(epoch) tan a r-ri ?• 



v h ; A cos e + A' cos e' 



Cor. 2. Any number of simple harmonic functions, of equal 

 period, added together, are equivalent to a single harmonic func- 



