of the Conditions of Maximum Efficiency. 37 



ment of E and F from these positions would result in no 

 alteration of the ratio in which E divides AF. 



But this is the characteristic of a maximum or minimum 



FE 

 value of such a ratio. Hence the maximum efficiency is -r-~; 



and it has before been proved that the triangle ADE is similar 

 to one in which the sides homologous to AD, DE are the 

 electromotive forces themselves. Thus, for maximum effi- 

 ciency, the phases of / must follow the phases of e by an 

 angle equal to (180° + ADE), or, which is the same thing, 

 precede them by (180°- ADE) or (tt— ADE). This, reduced 

 to time, is 



(tt-ADE) i>T ^ A_ ADE\ ^ 



2/T \ IT J 



DN 



It is clear that -r-pr is the ratio of/ to e. The maximum 



FE 



AD 



efficiency -r-^ exceeds this ; and it is seen to do so in virtue 



of the existence of a coefficient of self-induction, the absence 

 of which would cause D to coincide with B. 



Addendum. — Analytical Expressions. 

 The value of the maximum efficiency in symbols is 



,(l+ -cos/5/ 



T 1 + * cos /3 



where tan/3= 7^. The fraction — ; can be easily 



- + cos B 

 e 



shown to be greater than unity. And the angle ADE may 



"red, as follows : — 



ADE = (2 % -/3), 

 tan % = 



where 



sin/3 



e 



[ X is the angle DHA or BAH.] 

 Adapting to logarithms. 



