30 



Figure 19. — Resultant of two components of the same speed. 



Designating the common speed of the two components by a, the 

 constant length of the resultant radius by A^, and its initial angle 

 with CY (when ^=0) by a^, it follows that 



Ai cos {at-\-ai)-{-A2 cos {at-\-a2)=A^ cos (a^+^s) (31) 



It will be seen therefore that any two components having the same 

 speed unite into a component of the identical speed. 



53. Two components oj different speeds. — If two components have 

 different speeds, the faster of the two generating radii CPi or CP2 is 

 continuously gaining on the slower, the angle between these radii 

 progressively changes, the parallelogram CP1P3P2 steadily changes 

 its form, and the length of the resultant vector CP3, together with 

 its speed, varies with the time. The curve generated by the result- 

 ant vector takes various forms, depending on the amplitudes and 

 speeds of the components; but the periodic variations in the length 

 and speed of the resultant vector only need be here considered. 



54. Variation in the length of the resultant vector. — When the ampli- 

 tudes of the two components differ, A^ may be taken as the amphtude 

 of the major component, ai its speed; and A2 the amplitude of the 

 minor component. Let b be the algebraic difference between the 

 speeds of the components, so that a2=ai-\-b. The faster of the gener- 

 ating radii of the two components evidently will overtake and pass 

 periodically the slower. Taking for convenience the origin of time 

 at a moment when the generating radii coincide, the two components 

 then have the form 



yi=Ai cos ait 



2/2= A cos {ai-'rb)t 



(32) 



In figure 20, CPi and CP2 are the positions of the generating radii 

 of the two components at any time t, and CP3 the resultant. Since 

 the angle YCPi represents ait and the angle YCP2 represents (ai + 6)#, 

 the angle between the radii, P1CP2, is {ai-{-b)t—ait=bt. 



