I02 KINEMATICS. [189. 



189. The composition of two or more simple harmonic motions 

 in the same line can readily be effected, even when the compo- 

 nents differ in period. But the resultant motion is not simply 

 harmonic. 



Thus, for two components 



#2 = 02 cos(ft> 2 /-he 2 ), 



putting w 2 ^+e 2 = 1 ;+( 2 - &) i) / + e 2 = &) i / + i + ^ Sa 7 where 

 S=(< 2 (!)/+ 2 6! is the difference of phase at the time t y we 

 have for the resulting motion 



=a l cos 

 and treating this similarly as in Art. 184 we find 



x(a l -\- a 2 cos 8) cos (o>^ + ^) # 2 sin 8 sin 

 or putting a 1 -{-a 2 cos 8 = a cose, a% sin 8= a sine, 



x = a cos (!/ + ej + e), 

 where a 2 =a 1 2 + a% 2 + 2afy cos 8 and tane=^ 2 sin8/(a 1 +a 2 cos 



190. These formulae can be interpreted geometrically by 

 Fig. 47, similarly as in Art. 185. But as in the present case 

 the angle 8, and consequently the quantities a and e in the 

 expression for x, are variable, the parallelogram OP 1 PP 2 while 

 having constant sides has variable angles and changes its form 

 in the course of the motion. 



A mechanism similar to that of Fig. 48 (Art. 187), can be 

 used to effect mechanically the composition of simple harmonic 

 motions in the same line whether the periods be equal or not. 

 This is the principle of the tide-predicting machine devised by 

 Lord Kelvin.* 



* See THOMSON and TAIT, Natural philosophy, Vol. I., Part I., new edition, 1879, 

 p. 43 sq. and p. 479 sq. and J. D. EVERETT, Vibratory motion and sound, 1882. 



