Boever — Geometrical Properties of Lines of Force. 295 

 the subscripts 



r=±— =^^ (43d), 



|/C08 0) 



This equation represents the locus of the verticies of the 

 curves represented by equation (43a). 



Fig. 6 shows curves represented by the equations : 



r = Vq y^cos CO — 2 sin w, r = r^ ^/cos co -f- sin w, 

 and r = r^ i/cos «o, in which 01 = r^. 



Fig. 6 shows that all the lines of force which are outside the 

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

 points 



3. 



dx^ dx \ X m X 



) = 



67r<T (a;2 + y'^y /^-rra 



m x^ 



{-^{^^^fyy-^) = ^ (44) 



Therefore, 



71% 



y i/«2 + 2/2 = 2;— or yr = r^^ (45) 



or 



r-=±^=^ (45a). 



K cos CO 



Equation (45a) represents a curve which has two branches. 

 The plus branch is the locus of the points of inflection. The 

 curve is symmetrical with respect to the axis OY and the 

 branches are symmetrical with respect to the axis OX. For 



77" 



ft) = 0, ?' = =b Tg and for w = _ , r = ± 00. Hence the locus 



of the points of inflection passes through 1 and approaches the 

 axis OX as an asymptote. (Fig. 6.) 



Equation (45a) is the same as equation (43d). Hence the 

 locus of the points of inflection is the locus of the verticies of 



