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 8 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 8. The polar equation of this curve, 

 when referred to as a pole and Y as an initial line, is 



r 



= ±r l/cos&) — 8 sin co (43a). 



This equation represents a curve which has two loops, one 

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

 The -+■ loop alone has the property of cutting lines of force 

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

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

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

 1 is above O. For r = 0, cos co = 8 sin co or cot <b 1 = 8. 

 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 <w = — 8 cos co 

 or tan &> 2 = — 8. This shows that the longest radius vector 

 is perpendicular to the tangent at O. For a> = o> 2 + &>' 

 equation (43a) becomes 



r = ± r (1 + # 2 ) 4 i/cos «' (43b) 



i 



in which r 1 = r (1 + # 2 ) 4 is the longest radius vector. 

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

 and its longest radius vector as an initial line. Since 

 cos (+«')=cos( — a>') it follows that the longest radius 

 vector is an axis of symmetry. For 8 = either equation 

 (43a) or equation (43b) becomes 



± r 7/cos co (43c). 



r 2 



Since tan co = — 8, cos co = 7 ~ = _2_ or dropping 



V 1 -f 8 2 r. 2 



