123 



statical potentials (j, 2 , &c.), due to stated charges (q v q v &c.) 

 of the different wires, are expressed by the equations 



' m 



. r 



&c. &c. J 



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

 (or/ 1 ), w^, &c.) may be easily determined numerically to any required 

 degree of accuracy, in 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, 

 dv l dv. 2 dv 3 

 dx' dx' dx' 



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

 current at the same position, and k v k y k. A , &c. the resistances to 

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

 have by the law of Ohm, applied 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 



^i = _^y_i dq^___dy^ dh _dys ( >*\ 



dt dx' dt dx' dt dx 



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

 and then substftuting for v lt v 2 , v 3 , &c. their expressions (1), we 

 obtain 



_ dx k 2 dx k 3 dx 



which are the general equations of motion required. 



