626 PROCEEDINGS OP THE AMERICAN ACADEMY. 



magnitude of the force has sunk to zero. On the X axis, to the right 

 of this point, the force is in the direction BA ; to the left it has the 

 direction AB. 



Figure 4 is the same as Figure 2, except that the charges are of 

 opposite signs. The line of zero potential has the same shape in both 

 Figures 3 and 4. A curious survival of the tendency to the stationary 

 point is seen as we come up toward the origin from A. The equi- 

 potential lines are drawn in toward the origin, changing from concave to 

 convex. As in Figure 2, the lines of force were drawn free-hand. 



Figure 5 shows an elliptic and hyperbolic element, the parts in con- 

 tact being charged with the same sign. As in Figure 1, the overpowering 

 of one element by the other at the origin is to be remarked, the lines of 

 force springing from one charge to another of the same sign. 



Figure 6 is the same as Figure 5, except now the parts in contact 

 are charged oppositely. Here there are two stationary points at 

 (i> ¥) ' (h ~ i) » taking AB as the X axis and the other element as 

 the T axis. 



Figures 5 and 6 represent the two symmetrical combinations of 

 elliptic and hyperbolic elements. Figures 7 and 8 show the two 

 remaining unsymmetric combinations. It will appear that these fig- 

 ures contain nothing new; the separate quadrants have already appeared 

 in one of the other figures, but equipotential lines have now become 

 lines of force, and vice versa. Along the elements separating one 

 quadrant from another, discontinuities arise, which are explained by the 

 previous consideration of the discontinuities in </> and \p when regarded 

 as force functions. In what follows we shall abbreviate first, second, 

 third, and fourth quadrants by I, II, III, and IV. 



Figure 7 shows two hyperbolic elements. Except for the interchange 

 of lines of force and equipotential lines, I is the I of Figure 3, II the II 

 of Figure 1, III the III of Figure 3, and IV the IV of Figure 1 again. 

 The half stationary point is to be noticed, the transition from Figure 1 

 to Figure 3 coming across the line dividing this point. 



Figure 8 shows a hyperbolic and elliptic element. I is the I of 

 Figure 6, II of Figure 5, III of Figure 6, IV of Figure 5. The 

 potential here has only one stationary point, the halving of the num- 

 ber in Figure 6 corresponding to the halving of the single point of 

 Figure 3. 



