388 PROCEEDINGS OF THE AMERICAN ACADEMY, 



always be zero, and the sums of their projections on the —XX axis 

 will also always be zero. That is, at all times 



u — I -r. — ir = 0. volts (21) 



at ^ 



It is noteworthy that in Figure 8, both the discharging p. d. OUq 

 and the emf. of self-induction make an angle c^ with the current O/q, 

 the former leading and the latter lagging. Each of these emfs. there- 

 fore develops power on the current. This is a general condition of 

 the 0. c. circuit, different from that of the a. c. circuit, in which the 

 emf of self-induction exerts no dissipative power on the current, being 

 in quadrature therewith. 



We may, however, dispense with the equiangular spirals of Figure 8 

 by assuming that all the vectors rotate in circles with uniform angular 

 velocity w, provided we apply to their instantaneously projected values 

 on XOX, the proper damping factor e~-'' for the instant considered. 



The positions of the three vector emfs. and also their projections on 

 —XX, are indicated in Figure 8 for three successive instants angu- 

 larly separated by 30°, or 0.000,427,5 second, 



Returning to the stationary vectors of Figure 7, if we multiply the 

 w-diagram by the condenser-capacity c, we obtain the oscillatory 

 admittance diagram Y to a scale of mhos. The oscillatory admit- 

 tance is numerically the same as in the resistanceless case of Figure 3, 

 (0.006,324,6 mho), but makes a negative angle <^ with the reference 

 axis, instead of 90°. The oscillatory conductance is the real com- 

 ponent, and the oscillatory susceptance the —j component, of the 

 oscillatory admittance. 



Multiplying the Y diagram by f/o, the vector amplitude of the 

 initial discharging p. d., we obtain the current or / diagram, d ef, of 

 Figure 7. The initial vector oscillatory current I^ is 8.165 amperes, 

 corresponding to a r, m. s. initial oscillatory current / of 5.7735 

 amperes. The reactive component e f is the same as in the resist- 

 anceless case. The dissipative component d e is, the component in 

 phase with the discharging p. d. OUq Figure 8. A like component is 

 also in phase with the self-inductive emf OE; so that the total 

 equivalent component of dissipation current would be d d' Figure 7, or 

 10.328 amperes maximum cyclic initial value. 



U 



Multiplying the /diagram. Figure 7, by — ^, we obtain the stationary- 



vector power-diagram P, or the watts triangle g h k, which may be 

 drawn to any suitable scale of watts. This gives the maximum cyclic 



