342 BELL SYSTEM TECHNICAL JOURNAL 



the positive one being that sought. By expanding the quantity under 

 the radical according to the binomial theorem this root reduces to 



pX= lyf-rr Vk sin gr — k sin qr (44) 



when higher powers of k are dropped. 



The coefficients of the series then become, using the value of A 

 from equation (31) and the value of pA from equation (44), 



I: bo' = Mo<2 cos qr + 2p{2P -\- Q)Q cos qr -]-( t -- ) vPQ ^ sin qr 



L \ IT / p 



-f IttvQ^- sin qr (1 — cos qr) + vQ' sin- qr, 

 P 



= - 2ixoQ-sm qr - 8vP(3-?sin gr + — vP'^ 

 p p 37r 



= MoP + 2vP\ 



= HqQ - sm gr + — vPQ , sm gr, 

 P ^ P 



= -:^vPQ~sm qr, 



2 ^g . 8 p 



= 0. 



(45) 



The relation (29) can be used to return equations (45) to the general 

 time coordinate. Replacing r by / — X gives 



cos qT = cos gX cos qt + sin gX sin qt, 

 sin qr = cos gX sin qt — sin gX cos qt. 



As |X| never exceeds iv/p these equations can be simplified to 



cos qr — cos qt + . (2 sin p\ — sin 2p\) sin qt, 



^ (46) 



sin gr = sin qt — (2 sin ^X — sin 2/)X) cos qt, 



using the first terms of the F'ourier's series for sin gX in multiples of 

 p\. Upon substitution of equation (29) trigonometric functions of 

 ^X become the same functions of {pt — 2J7r), j an integer, since t is 

 defined as an even integral multiple of 2t/P. The phase angle is 



