9 g KINEMATICS. [183. 



beginning with those cases in which the resultant motion is 

 rectilinear. 



As, according to Hooke's law, the particles of elastic bodies, 

 after release from strain within the elastic limits, perform small 

 oscillations for which the acceleration is proportional to the 

 displacement from a middle position, the motions under discus- 

 sion find a wide application in the theories of elasticity, sound,, 

 light, and electricity, and form the basis of the general theory 

 of wave motion in an elastic medium. 



183. Two simple harmonic motions in the same line, of equal 

 amplitude a and equal period T, but differing in phase by &, com- 

 pound into a simple harmonic motion in the same line, of the- 

 same period T, but having the amplitude 2 a cos (8/2). 



For we have for the component displacements 



x^=a cos co/, x^ ft cos (fr>^+S) ; 



and as these are in the same line, they can be added algebrai- 

 cally giving the resultant displacement 



= #[cos o)/-hcos(i 

 or, by the formula cos a -f cos = 2 cos " ^ cos 



a 



x 2 a cos- cos &>/ + -. 



2 V 2 



184. Two simple harmonic motions in the same line, of equal 

 period T, but differing both in amplitude and in phase, compound 

 into a single simple harmonic motion in the same line and of the 

 same period. 



For the component displacements 



^ 1 = a l cos w/H-ej, ^ 2 = <z 2 cos 



can again be added algebraically, and the resultant displacement 

 is 



