582 



PROCEEDINGS OF THE AMERICAN ACADEMY. 



If the rotating vector be displaced 180° in phase, its projections will 



mark off instantaneous values of 

 energy, {iHj'l joules) in the in- 

 ductance. For reasons that will 

 be evident on considering oscil- 

 lating-current circuits containing 

 resistance, it is preferable to de- 

 scribe a circle upon a diameter 

 LHC, Figure 6, with the radius 

 In = Wjn, Figure 3, and rotate 

 this circle in the positive direc- 

 tion, with the angular velocity 

 2 CO, at the same time rolling it 

 along the axis OY, drawn through 

 the point L. The orthogonal 

 projection of the center S upon 

 Y will mark time, as indicated 

 in Figure 6 both to a scale of 

 seconds, and to degrees of energy 

 phase. The projection of the 

 point C in the circle on the OX 

 axis, commencing at <?, will mark 

 off a distance Oc corresponding 

 to the instantaneous energy 

 u^c/2 joules in the condenser. 

 Similarly, the projection of the 

 opposite point L in the circle on 

 the OX axis, commencing at /, 

 will mark off a distance 01 cor- 

 responding, on the same scale, to 

 the instantaneous energy iH/2 

 joules in the inductance. 



If we connect the points L and 

 C by a straight line, and take the 

 middle point, it will coincide, in 

 the resistanceless case here con- 

 sidered, with the center S of the 

 rolling circle. Consequently, the 

 projection of the point >S on the 

 OX axis at s, will mark off a 

 length Os, equal to half the sum of the instantaneous energy in the 

 condenser and in the inductance, i. e., equal to half the instantaneous 



S 2 



Joules Energy 



Figure 6. Rotating and rolling vec- 

 tor-diagram of the condenser energy, 

 the inductance energy, and the semi- 

 system energy in a resistanceless oscil- 

 lating-current circuit. 



