306 Professors Ayrton and Perry on the most Economical 
Let 6(v) be the watts per candie, expressed as a function of 
the number of volts employed at the terminals of a lamp; then 
H 
7a * PC) 
represents the cost per year per candle as far as the production 
of electric power is concerned. 
The total cost, therefore, per year per candle is 
xn H 
Foy x (6) + ag $() pounds, saga 
and we must find the value of v that makes this a minimum. 
There are two ways in which such a problem can be solved: 
the one a graphical method, the other an analytical method. 
The former may be used by even elementary students, and will 
be given first. It consists in drawing curves to represent, Lst, 
J(v) in terms of v, 2nd, O(v) in terms of v, and 3rd, d(v) in 
terms of v ; and from these the values of /(v), @(v), and o(v) 
are each determined graphically for many values of v, and 
the value of A calculated for each of these values. A fourth 
curve is then drawn, connecting the values of A with those 
of v, when it is easy to see by inspection for what value of » 
the expression A has a minimum value. 
The following is the result so obtained for the 108-volt 
Edison lamps used for lighting the Finsbury Technical 
College :— : 
p is taken at 5s., or £0°25 ; 
n as 560 hours, the time per year, approximately, during 
which the lamps are lighted ; 
H as £5: this is perhaps a rather high estimate for the 
cost of power, considering that the interest on the 
steam-engine, dynamos, &c., price of coal burnt, wages 
of the engine-driver and stoker have to be mainly 
debited to the driving of the College workshops, sup- 
plying power for the dynamos worked for experimental 
purposes; but itwill be accurate enough to take the sum 
of £5 per year as representing yearly interest on extra 
plant and the yearly interest on the extra cost of sup- 
plying one electric horse-power during the 560 hours. 
Then the value of v which makes A a minimum turns out 
to be about 106 volts; and on account of the flatness of the 
curve connecting the expression A with v, we see that in this 
particular case the annual cost of supplying light is only in- 
creased by 3°5 per cent., if the potential-difference between the 
mains be kept diminished to about 104°8 or kept up to about 
108 volts. Also, that keeping the potential-difference dimi- 
nished to about 1045, or increased to about 108°5 volts, 
increases our total annual cost of lighting by 5 per cent. 
