88 APPLICATION OF THE PRELIMINARY RESULTS 



the line above the differential coefficient indicating that it is to 

 be obtained by supposing the radius of dv to vanish after dif- 

 ferentiation, and this may differ from the one obtained by first 

 making the radius vanish, and afterwards differentiating the 

 resulting function of x, y , z , which last being represented as 



usual by -~r > we have 



the first integral being taken over the whole volume of A ex- 

 terior to dv'j and the second over the whole of A including dv. 

 Hence 



the last integral comprehending the volume of the spherical 

 particle dv' only, whose radius a is supposed to vanish after 

 differentiation. In order to effect the integration here indicated, 

 we may remark that X, Y and Z are sensibly constant within 

 dv, and may therefore be replaced by X t , Y t and Z t , their values 

 at the centre of the sphere dv, whose co-ordinates are x t , y t , z t \ 

 the required integral will thus become 



Making for a moment E = Xx + Y t y + Z t z, we shall have 

 X - Y- Z- 



and as also 



,1 ,1 



, d- , d- , 



x x r y y T z z 



~~^^~ 



,1 



, d- 



z z r 



