Potentials due to Moving Electric Charges. 133 



being the volume and surface densities, and the denominators 

 of the integrands having the same meanings as before. 



But if there is electricity at the point P either at rest or 



1 d 2 A 

 in motion, V 2 A — ^2^72 no l° n g er vanishes, and we must 



find what its value is. If we imagine a small sphere of 

 electricity removed, the expression becomes zero at P. 

 Hence its actual value at P is that due to* this sphere of 

 moving electricity. But we shall see that, if the radius 

 of the sphere diminishes without finite limit both A and 



~d 2 A 



■^Tjf tend to zero. Hence the value of the expression at P 



is the limiting value of V 2 A at P due to the sphere, when 

 its radius is made to diminish indefinitely. Inside this small 

 sphere the velocity of the electricity may be regarded as 

 uniform and as having the value that it has at P. 



Let an element of electricity, dq at Q at the time t be 

 moving with uniform velocity u in the straight line Q X Q 

 (fig. 2). Its companion point Q : is the position of dq at the 



P 



time t — , and the element contributed to the integral 



A by dq is 



dq dq dq 



PQ _p() UG0:i( f ) PQi — QQiCos^) PQcosa' 

 c 



But • r.^ 



sm a QQi u 



sin 6 PQx 



c 



