480 Prof. A. W. Riicker on the Self-Regulation 
efficiency as well as to the self-regulation, and have given the 
equations amore general form, in which they apply, when the 
symbols are properly interpreted, either to the external electro- 
motive force or to the external current. Dr. Frélich has 
dealt chiefly with the case of a constant electromotive force, 
in which the conditions of high efficiency and good self- 
regulation are in accord, and both can be secured together. 
To obtain the best result, if a constant current is required, 
will need a careful balancing of opposing requirements. 
As regards, however, the practical determination of another 
condition which shall render the problem determinate when a 
maximum value of @ does not occur between » and m, Dr. 
Frolich has advanced the question a step beyond where [I left 
it, and has added very greatly to the interest of his paper. 
He has shown that when the external electromotive force is 
in question, and without any reference to the value of the 
external resistance for which the maximum efficiency occurs, 
excellent results may be ebtained by taking (in the above 
notation) A°?Q=B?P. I had previously shown the physical 
meaning of this condition; geometrically it amounts to 
choosing the arbitrary constant, so that db /de=0 when «=0. 
Since, in the case under discussion, «=1/7,, Dr. Fr6- 
lich’s curves and mine are so related that the product of the 
abscissee of corresponding ordinates is unity, and the tangent 
to my curve at the point where it meets the axis of ¢ becomes 
in Dr. Frélich’s figures an asymptote. 
As expressed by the curve between e, and 7, (Dr. Frolich’s), 
good self-regulation can only be obtained in this case for 
values of 7, greater than that at which the curve may be con- 
sidered to have become practically parallel to its asymptote. 
As expressed by the curve between e, and 1/7, (my own), 
the corresponding condition is that shall be small enough 
for the curve to be at the corresponding point practically 
parallel to the tangent at the point for which w=0. Dr, 
Frolich has shown that these conditions are fulfilled in prac- 
tice, and has thus added very materially to the importance of 
his paper. 
It must, however, be noted that this solution has nothing 
absolute about it unless the efficiency is considered. If it 
is left out of account another constant (A) is indeterminate, 
and may be chosen so as to make ga minimum. For when 
TANG Ol) pyle 
_ em | AB(utm) + um(A+B) 
1~ (A+m)(B+m) AB+M(A+B) 
In the case under consideration we may put m=O, and thus 
