514 Mr. J. W. Peck on the Special 



If, however, the initial displacement conditions be related to 

 the initial velocity conditions so that 



arc cot ~ = m arc cot ~ .... (8) 



±m J-l 



for every harmonic, then (since the m disappears in equa- 

 tion (7)) the general rest epoch will occur; for evidently 

 among the rest epochs of each harmonic are found all those 

 of the fundamental. The first general rest epoch will occur 

 at time 



arc cot (t^T,) 

 2tt 



xT x , (9) 



and the subsequent ones at times increasing from this by 

 multiples of T/2. 



In the set of conditions 



arc cot ffr=i arc c °t rjf = i arc c °t rfr- = • ■ • • (10) 



J-l J-2 J- 3 



f 



it is to be noted that arc cot ~ is not to be taken as the 



smallest angle with the given cotangent. Suppose t x is 



given, then arc cot ± is to be chosen as follows : — If A 1 and 



B x are each positive, first quadrant ; if each negative, third 

 quadrant ; if A 1 positive and B x negative, fourth quadrant ; 

 if Ai negative and B x positive, second quadrant. If A m and 



B m are such that arc cot^ (formed in accordance with the 



±m 



above rule) is m times arc cot r -~ (formed in accordance with 



J-i 

 the above rule) for every m, then the necessary condition for 

 the rest epochs is satisfied. 



For the undisturbed epochs to occur we must have 



2?n7rt _ A m 'Irnir _ r m 

 ~T ~B, n "T~-~T^- 



or / t \ 



arc tan \-~ ) +kir 



L— 1 1,n ' nn 



T 2ir?n ' ' ' ' t 11 ) 



Hence, as before, these epochs will not occur in general 

 but will at times be given by 



arc tan | — .tjt) +kir 



•T„ . . . (l 2) 



