37^ COMPARISON OF RESISTANCES. 



as we approach the ends of the first dial, but by observing the 

 deflections of the galvanometer for the two contacts n and n + i 

 which comprise the position of equilibrium, we may, by a pro- 

 portion, find for a a more exact fractional value. 



977. MEASUREMENT OF VERY SMALL RESISTANCES. The use 

 of the bridge in its various forms serves very well for mean 

 resistances, but the method ceases to give good results whenever 

 very large or very small resistances are to be compared. 



The special difficulties presented by the measurement of very 

 small resistances are mainly due to the relative importance of the 

 resistances at the points of junction. On the other hand a 

 conductor with a very small resistance cannot be compared to a 

 linear wire, two dimensions of which may be neglected in com- 

 parison with the third.* 



We have defined above what is to be understood by resistance 

 of a conductor of three dimensions. Let us consider in the mass, 

 two infinitely small equipotential surfaces two small spheres, for 

 instance the one traversed by electricity which enters, and the 

 other by electricity which emerges; if these two spheres serving as 

 electrodes are respectively at the potentials Vj and V 2 , and that 

 I is the total current which traverses them, the resistance R is 

 given by the formula 



This resistance may be calculated by dividing the portion of the 

 medium comprised between the two surfaces of the electrodes into 

 tubes of flow, and applying to these tubes the properties of branch 

 currents. 



In order to measure the resistances of a conductor, four 

 electrodes are in general necessary : two of them, i and 4, put 

 the conductor on the path of a current ; two others, 2 and 3, 

 connect two points of the conductor with the galvanometer, or 

 any other measuring apparatus. Let V and I x with an index 

 equal to the number of the electrode, denote the potential and 

 the current for each of the electrodes ; we have, for the currents, 



(40) 



* KIRCHHOFF. Uber die Messung Elekt. Leititngsfahigkeit. Berliner 

 Monatsbericht, 1880 ; Gesammelte Abhandlungen, p. 66. 



