158 KENNELLY— THE LINEAR RESISTANCE [April 24, 



Ip 

 u = ~^Y abvolts (4^> 



To draw a stream line which shall include with the median line 

 OA the nth part of all the linear flux in the system, mark off on 

 OK a distance OG==a cot 27r/n; so that the angle OGA will con- 

 tain 2Tr/n radians. Then with center G and radius GA, describe 

 the circular arc AH, which is the required stream-line. 



It may be observed that if we draw two coordinate axes ov ozv 

 in the vzv plane, the function tanh {v-\-zv\/ — i) will correspond 

 on the 3;^ plane to the required loci, magnified by a. The locus 

 of this function, when v is given successive constant values and 

 zv alone varies, is a series of equipotential circles, while when zv 

 is successively assigned constant values and z' alone varies, the 

 loci of successive stream-lines are produced. If zv is expressed 

 in terms of -n- as ir/n and 2Z'= Y, we have 



OF = a tanh v^d — o- cm. (44) 



OB = a coth v^=d-\-(T cm. (45) 



CE = a/sinh Y = a cm. (46) 



OC=^a coth Y ^d cm. (47) 



also OH^a tan -rr/n cm. (48) 



OK=a cot ir/n cm. (49) 



GA = a/sm (2-ir/>i) cm. (50) 



OG^a cot 27r/n cm. (51) 



Fig. 8 presents the graphical construction of the function 

 tanh (v -\- wV — i) carried from the vzv plane to the ys plane, over 

 the limits v = — i to v=^-\-i and zi' = — 17/2 to zv= -{- Tr/2. 

 The points marked on the vzv plane have their corresponding points 

 marked on the yc plane. Thus the point p defined by ^'=I.o, 

 zv = 7r/2 on the vzv plane is represented by the point p defined by 

 y= 1. 313, .5 = 0, on the ys plane, or tanh (i -f- tt/^' V — i) = I-3I3' 

 Corresponding areas on the two planes are shaded alike. It fol- 

 lows from the formulas already discussed that linear resistances, con- 

 ductances and capacities are the same between corresponding conduct- 

 ing surfaces in the two diagrams. Thus, the linear resistance of the 

 double-cylinder system pqrs — tuvx is equal to the linear resistance 

 of the rectangular slab system with pqrs as one electrode and tuvx 



