370 Dr. J. A. Fleming on the Characteristic Curves 
certain variables which indicate as a most probable form an 
exponential function. 
There are four variables between which a relation is required 
—electromotive force, resistance, candle-power, and life. The 
third of these is at present somewhat vague and indeterminate. 
What we really are concerned with is the total eye-affecting 
radiation ; and our present methods only allow of a certain 
more or less imperfect comparison of this as a whole with that 
of a standard candle, or a calculation of the integral deduced 
from observation of the relative intensities of certain rays. 
Accordingly, of these four quantities two alone can be measured 
with any great accuracy. One is merely an average, and the 
other is necessarily a somewhat ill-defined quantity. 
Between any two of these variables we can seek a relation, 
and having a number of observations we plot down these in 
what are best called the characteristic curves of the lamp. 
Now two of the most important of these curves are those 
connecting the electromotive force and efficiency, or candles 
per horse-power, and the electromotive force and life; and they 
may therefore be called the principal characteristic curves of 
the lamp. Three other useful curves may be obtained by 
plotting down the curves connecting candle-power and current, 
candle-power and electromotive force, and electromotive force 
and resistance. These may be called subsidiary characteristic 
curves. 
Since the life and efficiency of a lamp vary together with 
the electromotive force, we can only properly represent the 
relation between the three by a surface, which may be called 
the characteristic surface of the lamp. 
Take three rectangular axes, «, y, z (fig. 1), and let dis- 
tances measured outwards represent life, candles per horse- 
power, and electromotive force. Leta curve be drawn on 
the y w plane, representing the relation of electromotive force 
and life, and one on the wz plane representing electromotive 
force and efficiency; let the ordinates be drawn at various 
points to both these curves, starting from abscissze, representing 
certain pressures v1, ve, &c. Complete the rectangles on the 
ordinates kh, 1, &c.; and we see that these rectangles form the 
orthogonal sections of a solid bounded respectively by the 
two planes «y and wz, and two curved surfaces, of which the 
characteristic curves / and k are the traces on these planes. 
The surface of this volume may be called the characteristic 
surface of the lamp. We see that the area of the orthogonal 
section parallel to yz gradually increases to a maximum, and 
then decreases. This is obviously because, for zero electro- 
motive force, life is infinite and candles per horse-power zero; 
