66 Mr. W. A. Price on Alternating 



Since each propagation is determined by a differential 

 equation of the same form as that which applies to a single 

 isolated conductor, such as is discussed in § 1, the action of Q 

 is equivalent to that of a pair of single conductors, and the 

 propagation along Q is the sum of the propagations along the 

 equivalent pair. 



Let the total resistances of the two equivalent cables be 27r/', 

 and 27rr 2 , and their capacities Iitc^ 27rc 2 . Then /j,)' 1 c i =l and 

 vr 2 c 2 =l. 



Since the pair is electrically equivalent to Q, they must 

 convey the same steady current as Q, when subject to a 

 steady E.M.F., and must be charged to the same potential by 

 a given charge. So ^"'-f r i ~ 1 ==r~ i , and ci + c- 2 ~c. From 

 these four equations r x r 2 c x c 3 can be determined. 



Then if a charge 27rS be introduced into this conductor at 

 any point, the charge at any point after a time t will be the 

 sum of the charges at the corresponding points of the 

 equivalent pair, if the same charge had been introduced into 



them in the proportions 27rS— , 27rS— 



If an alternating current 27rS cos cot be introduced into 

 the conductor at A where = 0, the charge at P will be 



Sc, s . n uP cos cot + co sin cot 



S = - - £ COS /6 C g- 2^— 



Sco 2 .^W 2 cos cot + co sin cot 



H i cos v#- 2 4 — = s l + s 2 . 



co c _ x co' + v r 



§ 4. Applying this argument to the conductors of the cable 

 described in § 2, let r\ r 2 c x c 2 be the resistances and capacities 

 of the pair equivalent to the central conductor, and r 1 V 2 / c/c 2 ' 

 of the pair equivalent to the concentric; so that 



' 1 + ? 2 — ? J ' 1 + '2 — * J 



r&fi = r 2 c 2 v = ViCi/x = r^cjv = 1 ; 



inn 

 Ci + e 2 = — — = c, c{ + c 2 ' = c' = n. 



m + ?i 



Then ^ == %' = _ifL_(l_ mj/r ) 



C C fJb — V 



(X—mfir), 



and SB = £L = 



c c' V — fl' 



Then the charge at any point P due to an alternating 

 current 2ttS coscot introduced at A, where = 0, is s=s 1 + s 



