2l8 VARIABLE STATE. 



Hence follows the conclusion which has been established above 

 (226), that the time required to produce a definite potential at the 

 distance x, or more precisely, a definite fraction of the potential at 

 the origin, is proportional to the square of the distance, and to the 

 coefficient a 2 , which is special to the wire. 



In these conditions, equation (2) really only contains one inde- 

 pendent variable, and putting 



IT 



it becomes 



20^ = 0; 



dz* dz 

 from this we easily deduce 



The constants A and B are determined by the initial conditions. 

 For = 0, that is to say x = or /= oo , we have V = V ; for = oo , 

 that is to say x = oo , or / = 0, we have V = 0. 

 We get then 



There is no simple expression for the integral contained in this 

 formula, but it is met with in a great number of problems ; for 

 instance, in the theory of probabilities, and tables of it have been 

 calculated, so that its numerical values are well known. 



JTT 

 Between the limits and oo , for instance, it is equal to > 



which gives 



