64 



Mr. W. A. Price on Alternating 



' These last expressions may also be obtained by expanding 

 (cos/30 + sm/30)eP in Fourier series of cosines. 

 For 



/3tt(cos fid- sin /3dy 9 



=5(-i)M(-W 



QO 



/37r(cos/30 + sin/30)^ e 



(-m{l+|4} 'cos^. 



h w} ] 



V 



{ 



COS? 



0. 



Putting 6 equal to and it successively in these series, and 

 adding and subtracting the series thus obtained in pairs, the 

 four last expressions are obtained. The method is applicable 

 only when /? is integral. 



In a paper by Dr. Grlaisher (' Proc. London Math. 



1 



Soc./ vol. vii.) the value of % 



r* + a* 



is determined by 



another method which gives the same result as the above. 



The physical significance of ft being integral is that in that 

 case the value of on is such that the alternations at the sending 

 and receiving ends of the cable are in the same phase. The 

 charge at the sending end being 



S 

 s=- fiir cosh fin (sin cot + cos tot) 



CO 



and at the receiving end 



S 

 s = - /3-7T cosech fiir (sin cot -\- cos cot) , 



CO 



Fig. 3. 



the ratio of the amplitudes 

 being cosh fiir. 



§ 2. A cable, shown in sec- 

 tion in fig. 3, contains two 

 conductors, an inner cen- 

 tral conductor and an outer 

 concentric conductor, insu- 

 lated from one another and 

 the water in which the 

 whole is immersed. The 

 whole cable is supposed to • 



be arranged in the circular ~ ZZZZZ^ZIZ^" 



form of fig. 2. 



2irr, 2irr are the total resistances of the inner and outer 

 conductors : 



