468 Prof. A. W. Riicker on the Sel/-Regulation 
external circuit—i. e. it is either the resistance or the conduc- 
tivity of the machine. If that resistance be represented by R.,, 
A=R,, if ¢ is the external current, and A=1/R,, if it is the 
external electromotive force. 
If « and m are respectively the largest and smallest values 
of « between which self-regulation is aimed at, then » —m may 
be called the range of w. There will in general be some par- 
ticular value of x which will be most frequently employed. 
This may be called the uswal value of 2, and may be denoted 
by & As it is sometimes necessary to distinguish between 
the cases in which # is a resistance and a conductivity, the 
value of 7, which corresponds to & will be indicated by 7». 
Conditions of Efficiency. 
Although a self-regulating machine is primarily intended to 
supply the same current or electromotive force whatever 
(within certain limits) the resistance of the external circuit 
may be, it is nevertheless desirable that it should be as 
efficient as possible. It is well known that the efficiency 
is a maximum for a given value of the external resistance ; 
and it is evident that the machine should not only havea high 
efficiency, but that the maximum should, if possible, be attained 
under the ordinary conditions of working. 
1. Short-Shunt Machine. 
Let 7 be the electrical efficiency. Then 
e /i? 2 ’ 
i) — SCN, poe died x 7 
19 ("2 + P1 + pa) 
ee Pa V9 
SS eeee—DHEe 
Totpitp,  (2+p1) (Tat py) + Tapa 
Taking the logarithms of both sides and differentiating with 
respect to 7, we get 
dicdy 1 1 ye Op 
Ges Meee ea 
dry % TetpytPa (72+) (Tat Pq) + Papa 
which vanishes if 
12(rq + Pa) = (p1 + Pa) 4p1(7at Pa) + Tapat 2 20 GR Ge 
Let 7’, be the positive root of this equation. Hence, since 
(72+ 1) (Ta+ pa) + %apa= (0’o+ Py) pat Taf 12+ pit pat ; 
and from (7), 
a {0/2 — (p1 + Pa)”} = pafpr(pr + pa)—7'3}, 
