466 



MI.MOIRS OF THE AMERICAN ACADEMY. 



In this equation (u) is the tension at any point, the distance of which from the middle 



of the circuit is (x) ; (a) is the tension at the point of 



(21) is the length of 



the circuit ; (k') is the conducting power, divided by a coefficient which expresses the 

 specific electrical capacity of the substance ; (t) is the time ; (it) is the ratio between the 

 circumference and diameter of a circle; (e) is the Napierian logarithmic base; and (i) 

 lb any positive number. M. Gangain, in his commentary upon the treatise of Ohm, 

 which he translated 1 into French, has pointed out the conclusions to be drawn from 

 this formula. He says that it has been established, on the assumption that there was 

 but a single electromotive force brought into play in the circuit, that this force was 

 const i nt, and that the circuit was homogeneous. In a voltaic pile there are many 

 electromotive forces developed at different points of the circuit ; if the resistance of 

 the pile is only a small fraction of the total resistance 



sensible error in the supposition that the pile is concentrated in one point, that its re- 

 sistance is nothing, and that the sum of the electromotive forces is represented by the 

 letter (a) of the formula. When the permanent state of the conductor is established, 



the value of (w) becomes equal to y^ x\; the term comprehended under the symbol (2) 



(which is obviously smaller as the time (t) increases) having become too insignificant 

 to bo regarded. This will happen as soon as (Tc n 2 ? t) is large, compared with (Z 2 ). 



of the circuit, there will be 



Whatever value of (t) may be sufficient for this purpose with any given value of (I) 

 it is evident that, in general, the required value of (t) must be proportional to (P) 

 This value of (/) represents the duration of the propagation before the permanent state 

 is acquired. It is evident that this value of (t) may be less as the value of (k r ) if 



greater. If the velocity of electricity is defined as ( ), it will have no determinate 



value, but may be exceedingly great or exceedingly small according to the distance to 

 be travelled. The passage of electricity is not analogous to the transmission of sound 

 or light, but resembles rather the conduction of heat. This will appear from compar- 

 ing Ohm's formula with that obtained by Poisson for the conduction of heat along a 

 metallic rod when the two extremities are maintained at a constant temperature. 2 



After Kirchhoff had succeeded in deducing the familiar formulas of Ohm, which ex- 

 press the constant voltaic current, from the principles of statical electricity, he gave 

 his attention to the variable state of the current, and he has obtained expressions 3 for 

 the quantity and intensity of the free electricity at any point of the conductor which 



1 Th^orie Mathematique des Courants Electriques, p. 177. 



1 Journal de TEcole Poly technique, XIX. p. 53. 



1 p °gg- Ann. der Physik und Chemie, C. 193 und CIL 529. 



7 



