56 DYNAMICAL THEORY OF SOUND 



with two complex arbitrary constants (7, G f . These can be 

 determined so as to identify z and z, at the instant t = 0, with 

 the vectors which represent the initial position and velocity of 

 the point (x, y). It appears from (14) that the most general 

 motion of a point subject to (13) may be obtained by the 

 superposition of two uniform circular motions in opposite direc- 

 tions. The same problem (virtually) has been treated in 18, 

 where the path was found to be an ellipse. This resolution 

 of an "elliptic harmonic" vibration into two circular vibrations 

 in opposite directions has important applications in Optics. 



The solution of the equation (11) may be derived from (14) 

 by taking the " real " part of both sides, i.e. by projecting the 

 motion on to the axis of x. Since <7, C' are of the forms 



C=A+iB, C'=A' + iB', (15) 



it might appear at first that the result would involve four 

 arbitrary constants. These occur, however, in such a way 

 that they are really equivalent only to two. Thus we find 



x = (A+A')cosvt-(B-B')smnt (16) 



The kinematical reason for this is that, as regards their 

 projections on a straight line, right-handed and left-handed 

 circular motions are indistinguishable. An important practical 

 corollary follows. We should have obtained equal generality, 

 so far as the solution of (11) is concerned, if we had contented 

 ourselves with either solution of (13), for example 



z=Ce int , (17) 



and taken the real part 



x A cosnt Bsmnt (18) 



This conclusion is obviously not restricted to the particular 

 differential equation (11) with which we started. The use of 

 an adjunct equation such as (12) has only been resorted to 

 in order to remove the suspicion of anything that can truly 

 be called " imaginary " in the work. Such assistance can 

 always be invoked mentally, but it is as unnecessary as it 

 would be tedious always formally to introduce it. If in any 

 case of a linear differential equation between x and t, with 

 constant real coefficients, we seek for a solution of the type 

 x = Cte xt , the imaginary values (if any) of X will occur in 



