﻿158 Mr. P. Cormack on Harmonic Analysis of 



Multiplying both sides by 6^ -0 ) gives 



e 2K<p~e)_ 1==ne 2i<i>_ ne -2ie t 



... e 2i (t-^{l~ne 2i9 \ = l- 



ne 



-2i9 



Mt-0) = (l-ne- 2ie )l{l-ne™). 



IT , 7T 



- and — y 



Since (p — lies between -f- and — ^, and n is less 



than unity, we may write 



2i(cj>-6) = log (l-^- 2 *' )-log (l-ne 2i9 ) 



= - ne -2i9_±_ n 2 e -ii9_i n 2 e -QzO_ t < t 



+ ng 2 ^ + \n-e^° -f- •fn 3 *? 6 *' + . . . 



= n. 2i sin 20+ ^n 2 . 2z'sin4(9 + §n 3 . 2i sin 6(9 + . . . 



... <£-# = w sin 26 + \n 2 sin 4(9 + l?z 3 sin 60 + (2) 



It will be evident that (2) gives the displacement of the 

 driven shaft relatively to the driving shaft. In practice 

 the angle between shafts joined by a Hooke's coupling 



rarely exceeds 15°. Since cos a = r , we have 



J 1 + n 



n — (l--co3a)/(l + cosa) = tan 2 -. 



For a=15° we get w = '0173, so that we can without 

 appreciable error neglect the terms containing the square 

 and higher powers of n in (2) and put 



$-d = 7i sin 26 (3) 



For the above value of a, the maximum value of </> — 6 

 given by (3) is "0173 radian or nearly one degree. 

 From (2) we have 



</> = + n sin 26 + \n 2 sin 4(9 + |n 3 sin 66 + (4) 



dfy = /^\ 2 ( _ 4?zsin 26-8n 2 sin4(9-12?i 3 sin 6(9-...). 



• • • W 



In obtaining (5) and (6) we assume the series formed by 

 the term-by-term derivative of the member on the right 

 in (4) and (5) to be convergent and to converge to the 

 differential coefficient of the member on the left. In 



