Theory of Faults in Cables. 165 



This equation (25) corresponds to curve 1, fig. 1 (p. 62), and 

 is well known. 



To find the limiting form of the arrival curve when # = 0. 



By (23), when z is finite, 



^ cos far . ax' - °l± 



v = 2EX : — sm-y-e T 



c-sma / 



from x f — to x'—--. The initial potential v between the 

 same limits is 



Etf' 4:Z 



Therefore 



v I l+4=z~ — cos iTT . ax r -.*** ,-„ N 



-=—• — — 2 : — sm T e t. . (26) 



v os zz a — sm a I v 7 



The (2i— l)th and 2ith terms are 



. a 2 i-\0s r _ a 2t-i* 



Z l + 4?/ sm_ - T~ 6 T 1 . 2W -^»\ 



- — — I -F-sm — =- 6 t 1 



A y 2z \ aa^-sm^.! 2?tt Z /> 



where a 2 i-i lies between (2i— l)7r and 2i7r, and ultimately 

 becomes 2wr when z is indefinitely reduced, so that the last 



expression takes the form -^. Evaluating in the usual manner, 



remembering that 



tan^*- 1 



2 

 2a 2i -i 



the (2z — l)th and 2ith. terms become 



2iiraf Unrlt . 2W\ _ ^l b 



T-—*7T sm -r) 6 T • 



Consequently (26) becomes, when 2=0, 



^Ufarlt . liirx' 2iirx'\ _J*E>? 



-( 



cos 



« ^{kfarlt . 2?W 2wne'\ 



— = 2z( — 77n-sm — y- cos — — ) 



T 



v . r 



Now, when a/ is indefinitely reduced, — is the same as =j- ; 



v o A o 



therefore, when x r = 0, 



r = ||^_ 2 | e -^ (27) 



From (27), curve 3, fig. 1, is calculated. The intermediate 

 curve 2, fig, 1, for which £=J, is calculated from equation 

 PHI. Mag. S. 5. Vol. 8. No. 47. .%; 1879. N 



