384 Curves and Surfaces of Incandescence Lamps. 
The first term is the expense of the translating device, and 
it mav be called the cost of lampage. 
The second term is the cost of the production of the energy 
which passes through the translating device or lamp-filament 
and is converted into eye-affecting radiant energy, and this 
may be called the machinage; hence the total expense of 
keeping going incandescent light is made up of lampage and 
machinage. Now, we can procure a given amount of light 
either by running the lamps very high, in which case lamps 
will cost a great deal, and power less in proportion, or we can 
run the lamps very low and save in lampage, whilst expending 
more in power in a greater number of lamps; and the question 
arises, apart from capital expenditure, at what point is the 
greatest economy obtained, or, in other words, what proportion 
ought lampage to bear to machinage in order that the total 
cost may be a minimum? To solve this we shall assume, as 
is very probable, that for the limit of variation of H.M.F. 
employed the average life of the lamp is an exponential 
function either of the candles per horse-power k, or of the 
candle’s light c, for the particular lamps considered. 
Let l= = anda — : be the functions. a@ and @ have defi- 
nite values at a given .M.F.; A and B are constants. 
1 1 
Then l= BIL ; k=Atl 
Substituting, we have 1 
Bipl + APL a=; 
in which B/ and A’ are constants. 
Now this expresses the total cost of working as a function 
of the average life at a certain H.M.F. 
Let us also take that las, where v is the H.M.F. at which 
the lamps are being run. ” 
Then 1 = 1 1 
Bes) +arP(5) <2 
ees a 
Blov 8B Y+A!Pv o=T. 
Let this be varied by varying v ; to find at which value it 
becomes a minimum with respect to v, differentiate with 
respect to v, and equate to zero. 
1 —pP ee Yy ay 
/ y BLES OP Res 
aT B a re? B A ate 
=== ee 
dv v ee: 
ait 1-Bs 
a : P Bynes 
or 
—a 
