tional area. To reduce the first cost we would, on the other hand, employ leads of small weight—that is, of small sectional area. We see that first cost and the subsequent working expenses are both governed by the area of conductor chosen, but whilst the former increases with the area, the latter decreases as the area increases, and it is evident that in each system of electric transmission of energy there must exist at least one particular area of conductor for which the sum of interest on its first cost, and annual cost of energy wasted, becomes a minimum. It is also evident that, notwithstanding any other consideration, this particular size of conductor must be adopted, being the cheapest in the long run.

The determination of this most economical size of conductor is somewhat complicated, and must be made specially for each case, regard being had to the following: 1. The rate of interest to be charged on capital outlay; 2. The cost of one horse-power-hour at the terminals of the generator; 3. The number of hours per annum that the maximum energy is required, and the number of hours that three-fourths, one half, and one quarter this amount is required; 4. The cost of unit weight of the conducting material; 5. The cost of insulation; 6. The cost of supports if an overhead line, or troughs if an underground line; 7. The cost of labour in laying. If it were permissible to consider the capital outlay as proportional to the total weight of conducting material, then for a given line we have the relation p K = k a p, where K is the total cost of the line, k a constant and p the annual rate of interest. The resistance of the line is inversely proportional to the area a, and the energy wasted equals resistance multiplied by the square of the current. Let q represent the cost of one electrical horse

power-hour at the terminals of the dynamo, and let t represent the number of hours per annum during which the current c is flowing-there being always the full amount of energy transmitted-then we have the annual value of energy wasted,

✓ w q t
p k

By inserting this value into the equations for K and W we find

Hence

pK= c√ wqt kp and

W = c √ w q t k p

p K = W,

or the most economical area of conductor, will be that for which the annual interest on capital outlay equals the annual cost of energy wasted. This law is commonly known as Sir William Thomson's law, and was first published by him in a paper on " The Economy of Metal Conductors of Electricity," read before the British Association in 1881. It should be remembered that this law in the form here given only applies to cases where the capital outlay is strictly proportional to the weight of metal contained in the conductor. In practice this is, however, seldom correct. If we have an underground cable, the cost of digging the trench and filling in again will be the same whether the cross-sectional area of the

cable be half a square inch or one square inch or more ; and other items, such as insulating material, are if not quite independent of the area, at least dependent in a lesser degree than assumed in the formula. In an overhead line we may vary the thickness of the wire within fairly wide limits without having to alter the number of supports, and thus there is here also a certain portion of the capital outlay which does not depend on the area of the conductor. It would, therefore, be more correct to write

K = Ko + ka,

where Ko represents that part of the capital outlay which is constant and independent of the area of the conductor. This addition on the right-hand side of the formula makes no alteration in the differential equation, for d Ko

but the value of p K is altered.

p K = p Ko + c √ wqt k p

Wcwqt k p.

The interest on capital outlay, and the annual cost of energy wasted are now in the relation

They are no longer equal, but the interest on capital out

lay must be greater than the annual cost of

By writing the above equation in the form

p (K— Ko) = W,

energy wasted.

we find that the most economical area of conductor is that

for which the annual cost of energy wasted is equal to the annual interest on that portion of the capital outlay which can be considered to be proportional to the weight of metal used.

Professor George Forbes, in his Cantor Lecture, on "The Distribution of Electricity," delivered at the Society of Arts, in 1885, called that portion of the capital outlay which is proportional to the weight of metal used, "The Cost of Laying One Additional Ton of Copper," and he showed that for a given rate of interest inclusive of depreciation, and a given cost of copper the most economical section of the conductor is independent of the electro-motive force and of the distance, and is proportional to the current. These facts can also be seen from the above formula

since the square root is a constant for each case, and since neither distance nor electro-motive force appear in the expression for a, which is simply proportional to c.

Having in a given system of electric transmission settled what current is to be used, we can, by the aid of Sir William Thomson's law, proceed to determine the most economical size of conductor. To do this we must know the annual cost of an electrical horse-power inclusive of interest and depreciation on the building, prime mover, and dynamo, we must know what is the cost of laying one additional ton of copper, and we must settle in our mind what interest and depreciation shall be charged to the line. These points will serve to determine the constants of our formulas, and then the calculation can easily be made. To avoid the labour of going through these

figures for every special case, Professor Forbes has prepared and published in his Cantor Lectures, some extremely useful tables which are reproduced on pages 185 and 186. Table A refers to the cross-sectional area of conductor required to carry a current of 1,000 amperes if the annual cost of one electrical horse-power varies from £5 to £20. Table B refers to the cost of laying one additional ton of copper, and interest and depreciation on it. The use of the tables will best be seen by an example. Say we have to transmit 50 amperes, the annual value of one horse-power is £10, and the cost of the line is £110 per ton of copper plus a constant. We shall also assume that it has been decided to charge 7%, for interest and depreciation on the line. We look in Table B horizontally along the line opposite 7 till we come to the vertical column headed £110. We find thus the figure 424. We now look in Table A horizontally along the line opposite £10 until we find again ·424 or the nearest figures to it. In the present case 441 and 411. The heading of the vertical column corresponding to this figure gives the area of conductor necessary for 1,000 amperes. We find thus that the conductor should be between 2.8 and 2.9 square inches-say average 2.85. But since our current is 50 and not 1,000 amperes, the area of conductor will have to be

50

1,000

× 2.85 1325

square inches. If we were to adopt a larger conductor the system would be less economical, because the capital outlay would become too great, and if we were to adopt a smaller conductor the system would be less economical because the waste of energy would be too great.