398 PROCEEDINGS OF THE AMERICAN ACADEMY. 



L\ — T^Z [^ = IZ \^ l^ = IJojo L^ Diax. cyclic volts (24) 



Frn = lU [^ = ^ /^ = ^^^M>. max. cyclic watts (25) 



2 ^ 



Or the discharging p. d. is ^° of current phase ahead of the current and 

 the undamped power reaches its maximum ^° of its phase ahead of 

 the current, all being considered as undamped. 



In order to derive a rotative energy vector-diagram, we take the 

 triangle Imn of Figure 7, or L'MN of Figure 12 and lay L' M along 

 the OY axis as shown in Figure 14. With the point iVas pole, the 

 equiangular spiral NL' is drawn, of angle ^. This will be tangent to 

 the Y axis at L' . We then draw a vector NC = NM and making 

 with NM an angle of 2</). The mid-point /S' of the straight line NO 

 is then connected to N by the vector N8. We may call NC the con- 

 denser-energy vector, iVxl/the reactive-energy vector, and ^'^>S'the semi- 

 system-energy vector ; i. e., the vector of the half sum of the energy in 

 the condenser and reactance. We now rotate the three vectors and 

 the spiral, with angular velocity 2a), while permitting the spiral to roll 

 along the axis Y. The successive turns of the spiral are to be capable 

 of rolling on this axis, as by displacing them infinitesimally out of the 

 plane of the paper, like the wards of a conical band spring. The vectors 

 iVC, N8, and NM are also to shrink as they rotate by application of 

 the damping-factor e~2x< Then the projections of C, J/, and /S', on the 

 OX axis, will define the instantaneous energy in the condenser, react- 

 ance and semi-system respectively. The path of the pole N will be the 

 straight line iVT, pursued with damped velocity. The paths C and M 

 will be exponential cycloids, that of >S' an exponential trochoid. The 

 positions of (7, >S', and M are traced in Figure 14 for several energy 

 phase intervals of 30°, or 0.000,213,8 second, the first two of which are 

 marked 1, 2, on each curve. It will be seen that taking the energy 

 scale along OX conformably with that of LMN, the condenser energy 

 starts at 2 joules, and after 60° or 0.000,427,6 second, it falls to 1.3811 

 joules. The reactive energy ol commences at zero and after 60° power 

 and energy phase it rises to 0.3544 joule. The semi-system energy Os 

 starts at 1 joule, and after 60° falls to 0.8678 joule. The displacement 

 V is therefore half the dissipated energy = 0.1322 joule. The total 

 dissipated energy at this instant is thus 2 s^s = 0.2645 joule. All three 

 vectors finally terminate and shrink into the point T. The distance 

 TL' — NL' /go^ <f), and NT is perpendicular to NL'. 



The fundamental differential equation for quantity q is satisfied by 



q - ^e-(^+-J''-)< -f Be-(^-J'^)\ coulombs (26) 



