PART II. POLAR MAGNETIC PHENOMENA AND TERKELLA EXPERIMENTS. CHAP. II. 435 



when: ill' is the magnetic force produced by the current-element at the place, d the distance from the 

 place to the element, and a the angle between this distance and the direction of the current-element. 



We now have to determine the force-components. The decomposition will be effected along the 

 radius vector, the meridian, and the parallel circle. 

 On looking at the figure we obtain 



x = Q sin 8 cos w 



v = Q sin 6 sin to (6) 



Z = Q COS 6 



Thus the direction-cosines for the radius vector are 



sin 6 cos ID, sin 6 sin 01, and cos 9. 



The direction-cosines of the tangent to the parallel circle are 



sin M, cos (a, and 0, 

 whence again we obtain the direction-cosines of the meridian, 



cos cos 10, cos 6 sin w, and sin 0, 

 which is immediately apparent on looking at the figure. 

 For the distance if, we find 



if- = L- -f- p 2 2Z.p [cos L cos 9 -|~ sin sin 6 cos (w <<)]. 

 The direction-cosines for this distance d are 



L sin cos fi Q sin 6 cos 10 L sin f sin p sin sin w , /. cos q cos 6 



j ~" ' j ~" ' <uid "" j 



it d d 



The direction of the force is now perpendicular to the current-element, of which the direction- 



cosines are 



sin /i, cos fi, 0, 



and to the direction towards the current-element. From this we find the cosines for the direction of 

 the magnetic force, 



L cos o cos 6 L cos 'C o cos 9 L sin a sin 9 cos (j /<) 



*-* - cos//, 2_lf Sln/ ,, and- g- 



where 



A = V(Z. cos C^p cos 0) 2 + [Z. sin C sin cos^w //)]- 



For a we find the following expression: 



, _ _^^_ A 



sin a = -^ V( cos cos 0) 2 + [ sin p sin cos (w p)J ^ 

 Hence we find 



and for the components 



= ,' sin ? sin cos - C S ~ L sin ^ cos fl 



,/P 9 = iL sin 'C. (L cos L" cos 0) "'fl + L sin - sin # 'Js* ' and 



dP,, = - iL sin ;- (Z cos e cos 0) -^i" ~ ^ ^ . 



If we put 



f" 



cos (j ft) an T "f 



M = ^s ' - ~ J* ' 



J/'O 



in which the lower limit may be chosen at pleasure, we obtain 



