of the Conditions of Maximum Efficiency. 33 



complete the right-angled triangle. For if ABO is such a 

 triangle, — AB, BO, AO representing respectively the preli- 

 minary resultant, the electromotive force of self-induction, 

 and the final resultant at the maximum values, — it is clear that 

 the maximum rate of increase of the resultant electromotive 

 force will be AC x angular velocity. Divide this by the 

 resistance, and the maximum rate of increase in the current is 

 obtained, which, multiplied by the coefficient of self-induction, 

 must give the maximum electromotive force of self-induction, 

 from the fundamental conception of that magnitude. 

 Hence, in symbols, if r = the resistance, 



L = the coefficient of self-induction, 

 co = the angular velocity, 



BC= — o>L, 

 r 



^ = tanBAC=^. 

 AO r 



If 2T is the period, co — -^, .-. tan BAC= 7^. 



And since the electromotive force of self-induction must be 

 greatest and +ve when the current is changing through zero 

 from + ve to — ve, it is clear that the phases of the electro- 

 motive force of self-induction must follow those of the final 

 resultant electromotive force at an interval of time represented 

 by a quarter of the period. Thus the above construction is 

 justified. 



6. The power working at any instant in a source of electro- 

 motive force is the value of the product of the instantaneous 

 electromotive force in question and of the instantaneous cur- 

 rent ; but this is constantly changing during a period, and 

 the mean power is half the product of the maximum value of 

 the electromotive force, of the maximum value of the current, 

 and of the cosine of the angle representing the time-interval 

 between their similar phases, I have given a geometrical 

 proof of this theorem in "Alternating Currents.'" It amounts 

 simply to this in the methods of representation here employed, 

 that if we project the revolving line corresponding to any 

 particular source of electromotive force upon the direction of 

 the final resultant, the power derived from this particular 

 source will be the product of such projection and the final 

 resultant divided by twice the resistance. Hence the various 

 powers of the different sources will simply be proportional to 

 the various projections upon the line of the final resultant. 



Phil. Mag. 8. 5. Vol. 25. No. 152. Jan. 1888. I) 



