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



or 



\o) — Trax = C , 



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

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

 force as flowing is correct. 



The slope of a line of force at any point (a;, ?/) is given by 

 equation (19), which written in another form is 



or 



r = 7*Q (cos (0 — S sin co) (20a). 



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



through (x = 0, y = 0) and / (x =0,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 O 

 is fS. For /S = equation (20) becomes 



/ 1 X \2 , _ 1 \2 



l2^~2^J"^''"-4 77^' 



which is the equation of a circle whose diameter is 01 = ^q ~ 

 — -. This circle cuts lines of force in points at which they 



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 

 dx^ 



= c7^K'-x(" + 2/^"~')) 



27rV2 



— r— 1/ (x^ + 2/^) — 27r<7 (x"^ +2/^) 



XS^ =0 (21)- 



