294 Trans. Acad. Sci. of St. Louis. 



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

 as equation (25). It shows that in this case also, the analogy 

 of considering force as flowing is correct. 



When /S is constant (43) is the equation of a curve which 

 cuts lines of force in points at which they (the lines of force) 

 have the same slope S. The polar equation of this curve, 

 when referred to O as a pole and OT'as an initial line, is 



r = ± J'qV^cos 0) — /S'sino) (43a). 



This equation represents a curve which has two loops, one 

 of which is represented by + and the other by — . (Fig. 6. ) 

 The -f loop alone has the property of cutting lines of force 

 in points at which they have the same slope. For « = 0, 

 r = zh r^. This shows that the plus loop cuts OY in /and 

 the minus loop cuts OY in a point which is as far below as 

 I is above O. For ?' = 0, cos co = /S sin to or cot oo^ ■= S. 

 This is the equation of a tangent at O. The curve is sym- 

 metrical with respect to this tangent. This tangent is parallel 

 to the tangents to lines of force at points in which they are 

 cut by the plus loop. When r is a maximum sin « = — /S cos w 

 or tan «2 ^= — ^- This shows that the longest radms vector 

 is perpendicular to the tangent at O. For (o =: co^ -{- co' 

 equation (43a) becomes 



?• = ± r^ (1 + /5')' i/cos <o' (43b) 



in which r^ = r^ (1 + /S'^)'' is the longest radius vector. 

 Equation (43b) represents the curve referred to O as a pole 

 and its longest radius vector as an initial line. Since 

 cos (+&)')= cos ( — ft)') it follows that the longest radius 

 vector is an axis of symmetry. For S = either equation 

 (43a) or equation (43b) becomes 



r = zt Vq |/cos (o (43c). 



1 ^2 



Since tan o, = — S, cos &>, = . = _L- or dropping 



