478 Prof. A. W. Riicker on the Self-Regulation 
which reduces to 
{(A—B)p+p—m\*=4p(A+m)(B+ p). 
Solving for B, we get 
Bp=Apt+ptm+2A+2/(At+p)A+m)1 +p). 
Hence, since A> B, Ap must be greater than the right-hand 
side of the equation. This inequality can only hold if we take 
the negative sign, and if 
IV (A+p)(A+m)(1+p)>(A+p)+(A+m), 
anes alike 
4p(A+p)(A+m)>(e—m)’; 
whence 
(4 —m)* 
PT h+py(Atiny 
Also, since B is positive, 
Apt+(A+p) +(A+m)>2/(At+p)\(At+m)(1+p); 
L+p—2/1+p/A+p)Atm)/A 
+(A+p)(A+m)/A?> pin / A’; 
{V1 tp—V (At p)y(Atm)/AS? > win /A?, 
We find, therefore, that in both these cases an inferior limit 
can be found to p or g, which depends only on pw, m, and A. 
If, therefore, the latter quantity is determined by considera- 
tions relative to the efficiency, we implicitly determine at the 
same time a limit to the perfection of the self-regulation. 
The last case is probably important practically, and the 
expression obtained shows that the inferior limit to p varies, 
if A is large, nearly as the square of the range. It will 
be less as A is greater; and this statement holds good for 
all the other cases enumerated above, so that a large value 
of A is favourable to good self-regulation. On referring to 
equations (8) and (10), we see that for a given usual value of 
« a high maximum efliciency is favourable to a large value of 
A if ¢ is the external electromotive force, and to a small value 
of A if ¢ is the external current. Hence we conclude that it 
is more difficult to combine a high efficiency with an approxi- 
mately constant external current than with an approximately 
constant external electromotive force. 
Postscript, April 30, 1885. 
Since the above paper was read before the Physical Society, 
