II4 KINEMATICS. [212. 



212. Let. a l = a z =a f e 1 = o, e 2 = &, and let the ratio of the 

 periods be T 1 /T 2 = 2/i. The equations of the component sim- 

 ple harmonic vibrations are 



x=acoso)t, y a cos(2a)t+S). 

 Here it is easy to eliminate t. We have 



y = a cos 2 tot cos B a sin 2 a>t sin S 



ni'y* i\rn*% o 



a \ ^ ~o L ^ Ub ^ ifr ~ \/ x o 



a* I a^ or 



Hence the equation of the path is : 



ay = (23? a 2 ) cos S 2 x^/cPx*' sin B. 



If there be no difference of phase between the components, 

 i.e. if S=o, this reduces to the equation of a parabola : 



For S = ?r/2, the equation also assumes a simple form : 



213. It is instructive to trace the resulting curves for a given 

 ratio of periods and for a series of successive differences of 

 phase (Lissajouss Curves}. 



Thus, in Fig. 51, the curve for 7\/ 7^ = 3/4, and for a phase 

 difference S = o is the fully drawn curve, while the dotted curve 

 represents the path for the same ratio of the periods when the 

 phase difference is one-twelfth of the smaller period. The 

 equations of the components are for the full curve 



3 4 



and for the dotted curve 



