472 Prof. A. W. Riicker on the Self-Regulation 
general equation in which both & and o are included, viz. 
(E—nk)(E+R/ocS)=nME /co. 
For if some of the quantities s,, s,, or s, are negative, and if 
the value of 7, is such that S=0, then H=nk. The approxi- 
mate expression which is being discussed fails therefore in this 
case, but it applies to cases in which one of the spirals is 
wound in the negative direction but in which B remains 
positive. Reference to the expressions for B shows that this 
will be the case in the Short-Shunt Machine if the series-coils 
are negative and if the number of turns is <s.p;/pa, and in 
the Long-Shunt Machine if the shunt-coils are negative and 
if ss<s,. In other cases Q is negative also. 
Confining our attention therefore to the case in which B is 
positive, let X be the critical value of #, and ® the corre- 
sponding value of ¢. 
Then 
KX=(AVQ—BYVP)/(VP—VQ), . . (GD 
b=(VP—VQ)/(A—B). .>. 9 a 
Also 
Lows aie 20 
dz ~ (A+a (B+a)>’ 
whence, at the critical point, 
a= —2(VP—VQ)'/(A—BYY PQ. . . (18) 
Hence if A>B, @ is positive and is a maximum; and if 
A<B, @ is negative and is a minimum. 
The value of X given by (11) is not necessarily positive, 
and hence we must consider a number of cases which differ 
from each other in the relative magnitudes of A, B, P,and Q. 
In distinguishing between them, it is convenient to remember 
that 
ree: a(P—Q)+BP—AQ_ 
, (A+2)(B+2) 
It follows from this expression and from (11) that 
(1.) If P/Q<A/B<1, 
@ is negative for all positive values of 2. 
CIT.) df A/G <P | @ als 
@ is positive for values of «<(BP—A(Q) /(Q—P). 
(JIL) If A/B<1</P/¥/Q, and therefore < P/Q, 
@ is positive for all positive values of z. 
