Boever — Geometrical Properties of Lines of Force. 283 

 Therefore 



y z=i — = ?• = 01, or r = — ?— 



TTCr cos 0) 



or the locus of the points of inflection is a straight line 

 parallel to AB and passing through 1. 



A circle througli O and / cuts each line of force which is 

 outside the critical line in two points at which the slopes are 

 the same. Hence the point of inflection must be between 

 these two points. As the two points approach each other 

 they approach the point of inflection, and reach it simul- 

 taneously. Therefore a circle through O and / is tangent to 

 some line of force at its point of inflection. 



If 8 is the slope of a tangent then 



(y, _ 1 _ dx _ \x 



/S dy ira {x^ ■\- y"^) — \y 



is the slope of a normal at the same point. The equation of 

 a normal at the point {x\ y') is 



\x' 



y — y' = TZnTTL — ?2l T~^ (^ — ^') (22). 



^ ^ 7ra {X ^ -\- y ^ ) — \y ^ ' ^ ^ 



-r , ^ 



If y' = — this equation becomes 



A, A,X , , , 



y =— H -, TT-^ — ^, {x — x). 



The intercept of this line on the axis Ol^is ?/ = 0. There- 

 fore normals to the lines of force at their points of inflection 

 pass through the origin O. 



If 3/' = equation (22) becomes 



y — r (x — x'\. 



The intercept of this line on the axis OT"is y = = — r,. 



