2o6.] PLANE MOTION. Iir 



Hence, to compound any number of simple harmonic motions 

 along lines lying in the same plane, we may project all these 

 motions on any two rectangular axes Ox, Oy taken in this 

 plane, and compound, by Art. 184 or 189, the components lying 

 in the same axis. It then only remains to compound the 

 two motions, one along Ox, the other along Oy, into a single 

 motion. 



206. Just as in Arts. 184, 189, we must distinguish two 

 cases : (a) when the given motions have all the same period, 

 and (f) when they have not. 



In the former case, by Art. 184, the two components 

 along Ox and Oy will have equal periods, i.e. they will be of the 

 form 



xa sin &>/, y = b sin (&>/+). (30) 



The path of the resulting motion is obtained by eliminating t 

 between these equations. We have 



^= sinft>cos-h cos wt sin 8 

 o 



x I *2 



=- cosS+\/i sin 8. 



/7 SI" 



Writing this equation in the form 



or 5-^ cosS+-^ = sin 2 S, (31) 



a* ab IP 



we see that it represents an ellipse (since -- - = [ ) 



\ ab J 



is positive) whose centre is at the origin. The resultant motion 

 is therefore called elliptic harmonic motion. 



