IIO KINEMATICS. [205' 



We assume the length of this hypotenuse =TT; then the given funcj 

 tion isf(x)=x from x=o to x=ir/2 } and /(X)=TT x from x=Tr/2 

 to X=TT. 



On account of the discontinuity at the point x=ir/2, the Integra 

 in (28) must be resolved into two, and we have 



2 r c^ c* ~\ 



B m = - I x sin mx dx + I (irx) sin mx dx 



7T \_J Jn J 



2 f I 7T MITT I . KlTT I TT #27T I . W7T~] 



= cos 1 5 sm 1 cos h 9 sin 



7T L m 2 2 #T 2 W 2 2 #T 2J 



For even values of m, sin(w7r/2)=o ; for odd values, sin(w7r/2) is 

 alternately positive or negative. Hence the series (23) becomes 



sin 



(29) 



This expansion certainly holds when x lies between o and ?r. A 

 every term of the series vanishes for x=o as well as for X=TT, th< 

 expansion holds even at these limits. Moreover, when x lies between i 

 and 2 TT, all the terms of the series, with signs reversed, pass through thi 

 same succession of values as between o and TT. The series represents 

 therefore between these limits an equal triangle with its vertex belo^ 

 the axis of x (Fig. 50). Beyond the point x=2-n, the same figure 

 repeats itself owing to the periodicity of the sine. 



It thus appears that the series represents an infinite zigzag line fo 

 all values of x. 



205. We proceed to the composition of simple harmonic motion* 

 not in the same line. We shall, however, assume that all tht 

 component motions lie in the same plane. 



It is evident that the projection of a simple harmonic motion 

 on any line is again a simple harmonic motion of the same 

 period and phase and with an amplitude equal to the projection 

 of the original amplitude. 



