412 PROCEEDINGS OF THE AMERICAN ACADEMY. 



Figure 18 shows the rotative vector-diagram of voltage and current for 

 the ultraperiodic case considered, and corresponding to Figure 8, the 

 rotative vector-diagram for the oscillating-current case. Lay down 

 the p. d. triangle i>A'i^ of Figure 17 at IqUqO, Figure 18, to voltage scale 

 as shown. On O/q as semi-axis, construct the rectangular hyperbola 

 HT^qTqUoH', whose asymptotes OA and 0^' make angles of 45° with 

 Ol^y From U^, draw a parallel to OIq, intersecting the hyperbola at 

 l\. Join Uq. Then the angle /qO 1/^ = ^ will be the gudermannian 

 of the hyperbolic sector 1^0 Uq ; or the hyperbolic sector angle TqOU^ 

 the anti-gudermannian of y\) . From the opposite corresponding point 

 E^ of the hyperbola, draw the straight line OEq. Then IT^ represents 

 the initial vector discharging p. d. in the_circuit considered, OA^ the 

 initial vector emf. of self-induction, and 01^ the initial vector discharg- 

 ing current, corresponding respectively to the vectors of the same 

 denomination in Figure 8. These three vectors, starting at the positions 

 shown, at the moment when the condenser of 4 microfarads, after being 

 charged at 1000 volts p. d., is released through /= 0.1 henry and 

 r =^ 500 ohms, run along the hyperbola HI^H\ in the positive direc- 

 tion, with uniform hyperbolic angular velocity O = 1936.492 hyps, 

 per second. That is, the sectorial areas described by each vector in 

 each second of time are constant and equal to 1936.492 hyperbolic 

 radians, taking the length of the semi-axis O/q as unity. Consequently, 

 at any instant, the sectorial areas EJ^I^ and 1^0 U^ remain equal to 

 that shown, this area being the hyperbolic angle of either, and equal 

 to 1.03172 hyps. = i/rt'.-^ 50°. 46'. 06". As the three vectors OE^, 

 OTq and OUq rotate with this uniform angular velocity in contact with 

 the hyperbola, they continually approach the asymptote OA', without 

 ever actually reaching it. 



The resultant, or vector sum of OU^ and OE^ is Or = 2582 volts, 

 and is equal to the initial vector product of the discharging current 

 O/q and the resistance r of the circuit. This vector also rotates with 

 uniform hyp. angular velocity fi, over the rectangular hyperbola h r h', 

 in the direction rh, corresponding to Or, Figure 8. As in Figure 8, the 

 negative of vector Or, or — I^ r, should be drawn in the direction 1^ ; 

 but is omitted from the diagram for economy of space. It may be 

 demonstrated that this negative extension of Or, rotating positively 

 over a rectangular hyperbola, the image of h r //, will always be in 

 vector equilibrium with O/i'o and OU^, so that the geometrical sum 

 of these three vectors at any instant will be zero, just as in the case 

 of Figure 8. 



