i8s-] 



PLANE MOTION. 



a 1 cos (wt + e^)+ tf 2 cos (at + e 2 ) 

 = (tfj cos ej-h <z 2 cos e 2 ) cos w/ (tfj sin ejH-^ sin e 2 ) sin at. 



Putting <2 1 cose 1 + ^ 2 cose 2 = ^cose, a sine^d^ sine 2 = sin e, 

 we have 



x=a cos e cos a>ta sin e sin a*t 



a cos (o)/+e), 

 where a 2 = (# j cos e 1 + ^ 2 cos e 2 ) 2 + (a^ sin 6 a 



= aj 2 H- 2 2 + 2 tf 1( z 2 cos (e a - e^ 

 and tan e= (a-^ sin e-^ + a^ sin e^)/(a 1 cos ej-f-tfg cos 6 2)- 



185. A geometrical illustration of the preceding proposition 

 is obtained by considering the uniform circular motions corre- 

 sponding to the simple harmonic motions (Fig. 47). 



sin e 2 ) 



Fig. 47. 



Drawin the radii 



= tf so as to include an angle 



equal to the difference of phase e 2 1 and completing the 

 parallelogram OP^PP^ it appears from the figure that the 

 diagonal OP of this parallelogram represents the resulting 

 amplitude a. For since P^P is equal and parallel to OP^ we 

 have for the projections on Ox the relation OPx^+OP* t =OP x , 

 or x^+x^=x. 



Again, if the angle xOP l be taken equal to the epoch-angle 

 e p and hence ^OP^ = e 2 , the angle xOP represents the epoch 

 e of the resulting motion. 



