112 



KINEMATICS. [207. 



207. Although in what precedes we have assumed the axes 

 at right angles to each other, this is not essential. The same 

 equation (31) is obtained for oblique axes Ox, Oy, and it is easy 

 to show (say by transforming (31) to rectangular axes) that this 

 equation still represents an ellipse. We have, therefore, the 

 general result that any number of simple harmonic motions .of 

 the same period and in the same plane, whatever may be their 

 directions, amplitudes, and phases, compound into a single elliptic 

 harmonic motion. 



208. A few particular cases may be noticed. The equation 

 (31) will represent a (double) straight line, and hence the elliptic 

 vibration will degenerate into a simple harmonic vibration, 

 whenever sin 2 8 = o, i.e. when 8 = n7r, where n is a positive or 

 negative integer. In this case cosS is +i or i, and (31) 

 reduces to 



- ^ = 0, if S = 



a b 



and to 



Thus two rectangular vibrations of the same period compound 

 into a simple harmonic vibration when they differ in phase by 

 an integral multiple of TT, that is when one lags behind the 

 other by half a wave length. 



209. Again, the ellipse (31) reduces to a circle only when 

 cosS=o, i.e. $=(2n+i)7r/2, and in addition a = b, the co-ordi- 

 nates being assumed orthogonal. 



Thus two rectangular vibrations of equal period and ampli- 

 tude compound into a circular vibration if they differ in phase 

 by 7T/2, i.e. if one is retarded behind the other by a quarter of 

 a wave length. 



This circular harmonic motion is evidently nothing but uni- 

 form motion in a circle; and we have seen in Art. 172 that, 

 conversely, uniform circular motion can be resolved into two 



