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 O as a pole and 0!Fas an initial line, is 



r 



=: ± ?oV^cos &> — 8 sin a> (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 &> = 0, 

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

 the minus loop cuts 03^ in a point which is as far below as 

 1 is above 0. For r = 0, cos (o = 8 &\x\ (o or cot w^ := 8. 

 This is the equation of a tangent at 0. 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 o> = — 8 cos a> 

 or tan <>>2 = — 8. This shows that the longest radms vector 

 is perpendicular to the tangent at 0. For a> = (o^ -\- a>' 

 equation (43a) becomes 



r = ±rj, (1 + /S^)^ l/coso)' (43b) 



1 



in which r^ = r^ (1 + 8^)^ 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 ( — a>') it follows that the longest radius 

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

 (43a) or equation (43b) becomes 



db r^ |/cosG> (43c). 



1 



Since tan »„ = — 8, cos w, = . u = -l- or dropping 



' * l/l + /S^ r/ ^^ * 



