414 PROCEEDINGS OF THE AMERICAN ACADEMY. 



considered ; or, we may shrink the vectors by the application of the 

 damping factor before taking projections, as in Figure 8. In the case 

 considered, the damping coefficient -l = 2500, as in Figure 17 ; so that 

 applying the damping factor e"'^^'^''', we obtain the curved lines of Figure 

 IS. The dotted line o3'2'lV is drawn as though with negative rotation 

 of Or, to simulate the projective effect of a negative vector — /^ r. 



The points 1, 2, and 3 on the hyperbolas, indicate the positions of 

 the various vectors after the lapse of 1, 2 and 3 ten-thousandths of 

 a second respectively, after release. The corresponding points 1', 2', 

 3' on the curved lines, give the termini of the same vector as reduced 

 by damping, and the projections of the latter, 1, 2, 3, on the XX 

 axis, give the corresponding instantaneous values in the circuit of the 

 discharging p. d. u, the ir drop, the current, and the emf of self- 

 induction, in the same manner as in Figure 8. It will be observed that 

 while the undamped vectors all increase in length without limit, the 

 actually projected values under the dominating influence of the damp- 

 ing factor, diminish in time without limit. In particular, the current 

 / reaches a maximum when the vector 01^ has swept over 1.0317 hyp. 

 radians, in a time 1.0317/1936.492 = 0.000^532,8 second. At the 

 same instant, the self-inductive emf vector OE^ will have reached the 

 transverse axis 01^, and its projection on XX will momentarily 

 vanish. Consequently, there will be no emf of self-induction in the 

 circuit at this instant ; because the current is stationary for that in- 

 stant, being about to diminish. After this instant, the self-inductive 

 emf changes sign, and propels the current along with the discharging 

 p. d. 



It will be noticed that as in the oscillatory-current case, both the 

 discharging voltage and the voltage of self-induction exert dissipative 

 activity upon the discharging current. 



It may also be noticed that although the orthogonally-projecting 

 rotating-crank vector-diagrams used in this paper are convenient and 

 useful devices for representing the actions in o. c. circuits, the polar- 

 coordinate vector-diagram, sometimes called the " time " diagram, is 

 not so well adapted for this purpose. The undamped vector quanti- 

 ties in the periodic case may indeed be represented by circles on the 

 polar-coordinate diagram ; but the corresponding damped quantities 

 are represented by spirals that are not so easily interpreted as equian- 

 gular spirals. Moreover, in the ultraperiodic case, the hyperbolic ana- 

 logue is missing in the polar-coordinate diagram ; so that apparently 

 there is no analogy presented in polar-coordinate representation be- 

 tween the periodic and ultraperiodic cases. It would seem, therefore, 

 that the orthogonally projected "crank-diagram" or "clock-diagram" 



