456 OHM ON THE GALVANIC CIRCUIT. 



The electroscopic force, at any place of such a prismatic 

 body, may be deduced from the differential equation («) found 

 in § 11. For this purpose we have only to integrate it, and to 

 determine, in accordance with the other conditions of the pro- 

 blem, the arbitrary functions or constants entering into the in- 

 tegral. This matter is, however, generally very much facili- 

 tated, with respect to our subject, by omitting one or even two 

 members, according to the nature of the subject, from the equa- 

 tion («). Thus nearly all galvanic actions are such that the 

 phaenomena are permanent and invariable immediately at their 

 origin. In this case, therefore, the electroscopic force is inde- 

 pendent of time, consequently the equation (a) passes into 

 d^^u be 



Moreover, the surrounding atmosphere has (as we have abeady 

 noticed in § 9.) in most cases no influence on the electric na- 

 ture of the galvanic circuit; then i = o, by which the last equa- 

 tion is converted into 



' = -dV-' 

 But the integral of this last equation is 



u =fx + c, (c) 



where f and c represent any constants remaining to be deter- 

 mined. The equation (c) consequently expresses the law of 

 electrical diffusion, in a homogeneous prismatic conductor, in all 

 cases where the abduction by the air is insensible, and the 

 action no longer varies with time. As these circumstances in 

 reality most frequently accompany the galvanic circuit, we shall 

 on that account dwell longest upon them. 



We are enabled to determine one of the constants by the.] 

 tension occurring at the extremities of the conductor, which has 

 to be regarded as invariable and given in each case. If, for 

 instance, we imagine the origin of the abscissae anywhere in 

 the axis of the body, and designate the abscissa belonging to 

 one of its ends by ^j, then the electroscopic force there situated 

 is, according to the equation (c), i 



/^i + c ; 



in the same way we obtain for the electroscopic force of the! 

 other extremity, when we represent its abscissa by x^, 



