282 Trans. Acad. Sci. of St. Louis. 



or 



\(jo — Trax z= G, 



in which C is the constant of integration. This is the same 

 as equation (1). It shows that the analogy of considering 

 force as flowing is correct. 



The slope of a line of force at any point (x, y) is given by 

 equation (19), which written in another form is 



or 



r = r^ (cos Q) — S sin w) (20a). 



If /S' is a constant this is the equation of a circle which passes 



through (x = 0,y = 0) and / (x =zO,y = —). This 



shows that a circle passing through and /cuts lines of force 

 in points at which they (the lines of force) have the same 

 slope jS. The slope of a tangent to this circle at the point 

 is /S'. For jS = equation (20) becomes 





which is the equation of a circle whose diameter is 01 = »'o = 

 This circle cuts lines of force in points at which they 



'TTff 



have no slope. Fig. 3 shows several of the circles. The 

 perpendicular bisector of 01 is the locus of the centres of all 

 the circles. 



Fig. 3 shows that all the lines of force which are outside the 

 critical line have points of inflection. For the locus of such 

 points of inflection 



d'^X d { -1 7r<7 / , 2 -1 \ \ 



27r2<T2 



-^^ y {x^ + 2/^) — 27r<T (a;' + y') 

 x5^ =0 (21). 



