Prof. Thomson on Peristaltic Induction of Electric Currents . 137 



&c. &c. J 



If the sections of all the conductors are circular, these coefficients 

 (cr/i^ OTjt^;^ &c.) may be easily determined immerically to any required 

 degree of accuracy, iu each particular case, by the method of electro- 

 statical images. The electromotive force per unit of length at the 

 position X will be, in the different wires, 



rfUj rfl»2 rfVg 



dx dx dx 



respectively, and therefore if y^ 73, 73, &c. denote the strength of 

 current at the same position, and k^, k^, k.^, &c. the resistances to 

 conduction per unit of length in the different wires respectively, we 

 have by the law of Ohm, apphed to the action of peristaltic electro- 

 motive force, 



Now unless the strength of current be uniform along any one of the 

 wires, the charge of electricity will experience accumulation or dimi- 

 nution in any part of it by either more or less electricity flowing in 

 on one side than out on the other ; and the mathematical expression 

 of these circumstances is clearly 



dqi-._dri^ ^:=_^, ^ — —^ , . . (3), 

 dt dx ' dt dx' dt dx 



Using in these equations the values of y-^, y^, 73, &c. given by (2), 

 and then substituting for v-^, v^, v^, &c. their expressions (1), we 

 obtain 



dq,_d ^\ rf(t!r,"'y,) ^ 1 d{ ,'^'y,) ^ 1 ^(t^.'^'ya )^.;^. \ 

 dt dx\k^' dx k., dx k^ dx J 



dx kj dx J 



(4), 



'^'?g)_^l^WM+&c 

 "•"A, dx 



which are the general equations of motion required. 



It is to be observed that k^, k„, &c., ra-/'), cr/^), m^W, &c. will be 

 functions of x if the section of the conducting system is hetero- 

 geneous in different positions along it ; but in all cases in which each 

 conductor is uniform, and uniformly situated witli reference to the 

 others along the whole length, these coefficients will be constant, and 

 the equations become reduced to 



