TO THE THEORY OF MAGNETISM. 109 



the integral being taken from x \ to x= + \. The total 

 value of Y' is therefore 



27T V X - x' + a 2 + X - a' 



_2,r^[y{(X + a0 2 + a 2 }--X- 

 'dxd z X 



_ 2 Cdx 

 J(r) 



the limits of the integral being the same as before. If now we 

 substitute for (r) its value *J{(x x'Y + af] we shall have 



dxd*X f dx d*X 



both integrals extending from x = \tox = + \. 



On account of the smallness of a, the elements of the last in- 

 tegral where x is nearly equal to x are very great compared with 

 the others, and therefore the approximate value of the expression 

 just given, will be 



* A d*X' , . f dx Ol 2/4 



iraA j-fi where A = / ^ - ^- t ^ = 2 log 

 dx 2 J *J{(x x) +} a 



very nearly ; the two limits of the integral being p and + p, 

 and p so chosen that when p is situate anywhere on the wire's 

 axis, except in the immediate vicinity of either end, the approxi- 

 mate shall differ very little from the true value, which may in 

 every case be done without difficulty. Having thus, by substi- 

 tution, a value of Y' free from the sign of integration, the value 

 of Y is given by merely changing x into x and X ' into X ; in 

 this way 



, . 

 dx 



