Eoever — Geometrical Properties of Lines of Force. 283 

 Therefore 



y =— - r^= 01, or r = 



ira cos 00 



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

 parallel to AB and passing through 1. 



A circle through 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 S is the slope of a tangent then 



fy, 1 dx \x 



o = 



/J:? dy 'Tra{x' -\- y^) — \y 



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

 a normal at the point (as', y') is 



\x' 



y — y' = . ,, , ,■> , r-7- (« — »;') (22). 



^ ^ ira (x^ -\- y^) — Xy ^ f \ j 



X 

 If y' = — this equation becomes 



2/= A 4. ^^ - {x — 7^). 



TTff ( X ' -\ —5- I 



\ ^ TtV / ITff 



The intercept of this line on the axis OY is y = 0. There- 

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

 pass through the origin O. 



If y' = equation (22) becomes 



X 

 y = ; (x — a;'). 



X 

 The intercept of this line on the axis Ol^is y = = — r.. 



