188 Mr. T. Gray on the Size of Conductors 
The expression («) has a minimum value when 
a = g VW =c \/§' ■ ■ ■ (1) 
where i is the fraction of the original cost to be allowed per 
second for interest and depreciation. 
Equation (1) shows that, from a purely economical point of 
view, the section of the conductor should be directly propor- 
tional to the quantity C, which, when the current is continuous 
and of constant value, is the current-strength in C.G.S. units, 
but which, when the current is variable, has such a value that 
the heat which would be generated by a continuous and con- 
stant current of that value is equal to the actual heat gene- 
rated. The section should also be directly proportional to the 
square root of the product of the cost of energy and the spe- 
cific resistance of the conductor, and inversely proportional to 
the square root of the product of the first cost and the rate 
of interest and depreciation to be allowed, while it is inde- 
pendent of the length of the circuit. 
Equation (1) gives in all cases an inferior limit below 
which it is not advisable to reduce the section of the con- 
ductor. It should be noticed, however, that the investigation 
proceeds on the assumption that the whole of the energy wasted 
is due to the heat generated in the main conductor by the 
current. This assumption may be, in many cases, inadmis- 
sible, owing to the varying E.M.F. at which the electricity 
would be supplied ; and in such cases the economy problem 
assumes different forms depending on the conditions imposed. 
Practical Example. 
Let us suppose, for the sake of illustration, that the cost of 
one erg, or E, is, reckoned in pounds sterling, 
F _ 1 
10 9 T' 
that the conductors are of copper having a specific resistance 
S = 1700, 
that the price of copper conductors manufactured is 
1200' 
and that the rate of interest and depreciation is 
1=7*5 per cent. 
