IO8 PROPERTIES OF CIRCULAR CURRENTS. 



lines of force, the other the intersection of the plane of the figure by 

 the equipotential surfaces. 



The action at the point O is 



If the circles are replaced by coils with rectangular section of 

 dimensions zb and 2<r, each containing n windings, and we confine 

 ourselves to terms of the second order, we have 



the term with c 2 would disappear as having the factof a 1 - ^x 2 . 



If we push the approximation as far as terms of the fourth 

 order, we find that the term in y 2 contains the factor (31^ -36^); 

 this factor may be make to disappear if we choose the section so 

 that 



b 6 



- = = 1-079. 



c V3 1 



There only remains then for the fourth order the term in y*, the 

 value of which is 



54 y 



125 4 



750. COIL WITH FOUR CIRCLES. We obtain a more complete 

 solution by means of four circular currents having the same axis, 

 symmetrical in pairs in respect of a point of this axis. We shall 

 assume that the currents are four parallel circles of the same sphere, 

 the centre of which is the point in question. Let a be the radius of 

 the sphere, r and r' the radii of the circles, x and x' their distances 

 from the centre, / the ratio of the number of the windings of the 

 great circles to that of the small. If the currents are in the same 

 direction, the horizontal components add themselves. We may 

 arrange two of the three indeterminates r, r and /, so as to nullify 

 the two first terms of correction in the value of x. As we have u = a 

 for all the currents, this condition is 



pr 2 (4*2 - r 2 ) + r' 2 (4*2 - x 2 ) = 0. 



pr 2 (^ - 1 2r 2 x 2 + 8* 4 ) + r' 2 (r' - 1 2r' 2 x' 2 + 8*' 4 ) = 0. 



