Theory of Terrestrial Magnetism. 403 



p, v are the direction-cosines of a line at right angles to PS 

 and ST, 



_ nl 



" PSx/F+^T 2 ' 



mn 



^~ psv 7 F+^ r? 

 -( m »+p) 



V PSv^ 2 + m 2 ' 



where the negative sign must be given to the root. Now if 

 F is the force at P due to the charge of surface-density cr on 

 the elementary area dd . sin 6 . d<f> at S moving relatively to P, 

 we know that it must be at right angles to PS and ST, and 

 equal to 



<rsmd.dd.dcl)X sin PSTyV + t 2 

 PS 2 



Hence, if SX, SY, BZ be the resolved portions of the force F 

 parallel to the axes of coordinates, and if dS stands for 

 sin . d6 . d(p, we have 



8X-. 



a . c/S . \/u 2 + 1 2 



PS 2 

 and similarly for SY, 8Z. 



Hence, if J J stands for the summation over the whole sur- 

 face of the sphere, and if X is the total force at P in the direc- 

 tion of the axis of a, 



'adSw^P + ^n 2 nl 



-ft 



PS 2 FSy/P + m* 



aw <iS / 



n 



j PS 2 PS 

 Similarly 



Z: 



-If 



PS 2 PS 



dS m 2 + P n 



PS 2 n PS 



Now the resolved part of the force along the axis of x is the 

 same as would be the force in that direction due to a distribu- 

 tion of attracting matter of density —naiv over the surface of 

 the sphere. Similarly the resolved part of the force along the 

 axis of y is the same as would be the force in that direction 

 due to a distribution of attracting matter of density —naic over 



