2ii.] PLANE MOTION. II3 



rectangular simple harmonic vibrations of equal period and 

 amplitude, but differing in phase by 7r/2. 



The results of Arts. 205-209 can also be established by 

 purely geometrical methods of an elementary character.* 



210. It remains to consider the case when the given simple 

 harmonic motions do not all have the same period. It follows 

 from Art. 189 that in this case, if we again project the given 

 motions on two rectangular axes Ox, Oy, the resulting motions 

 along Ox, Oy are in general not simply harmonic. 



The elimination of t between the expressions for x and y may 

 present difficulties. But, of course, the curve can always be 

 traced by points, graphically. 



We shall here consider only the case when the motions along 

 Ox and Oy are simply harmonic. 



211. If two simple harmonic motions along the rectangular 

 directions Ox, Oy, viz. : 



x=a cos - 



of different amplitudes, phases, and periods are to be com- 

 pounded, the resulting motion will be confined within a rec- 

 tangle whose sides are 2a p 2a 2 , since these are the maximum 

 values of 2x and 2y. 



The path of the moving point will be a closed curve only when 

 the quotient T^T 2 is a commensurable number, say = m/n, 

 where m is prime to n. The x co-ordinate of the curve will 

 have m maxima, the y co-ordinate n, and the whole curve will 

 be traversed after m vibrations along Ox and n along Oy. 



The formation of the resulting curve will best be understood 

 from the following example. 



* See, for instance, J. G. MACGREGOR, An elementary treatise on kinematics and 

 dynamics, London, Macmillan, 1887, pp. 115 sq. 



PART I 8 



