magnetic force, is in proportion to the square root of the exterior resistance of the circuit, we may conclude that if we increase the E. M. F. in proportion as the resistance of the circuit increases, we may maintain the same diameter for the electro-magnet, and the same force. Consequently, if, instead of introducing our standard electro-magnet of 1 centimetre in diameter, on a circuit of 100 kilometres, we introduce it on a circuit of 400 kilometres with a double number of elements, nearly the same effect will be produced. But the size of the wire must be changed, for this varies for a like diameter of electro-magnet in the inverse ratio of the of the resistance of the exterior circuit. This diameter, then, the wire of the standard electro-magnet being 1583 millimetre, must be, in the present example,

as 100000 400000, or as 17.8: 25.1,

:

which makes it 112 millimetre. We see by this that we may without inconvenience maintain for telegraphic relays the allotted dimensions.

We have said above that by doubling the number of elements in the battery we should obtain nearly the same force on a circuit of 400 kilometres as on that of 100 kilometres. This word "nearly " must not be overlooked, for the resistance of the circuit, in the example we have cited, is not 400 kilometres, but 437240 metres, and this number is not exactly four times 118620, which represents the resistance in metres of the first circuit; it is less, and consequently, to obtain exactly the same effect in both

cases, the battery of 20 elements must be increased to 38 4 elements, or 39 instead of 40.

We will now employ two electro-magnets of small dimensions, each having a resistance of 2200 metres, wound with No. 16 wire (44 millimetre), and connected direct to a battery of eight Daniell cells arranged for tension. Experiment shows that the force obtained from each of them will be 70 grammes at a distance of 1 millimetre. Consequently the total force developed by both will be 140 grammes. Now if, instead of employing two electro-magnets we use but one, this force will be 200 grammes. Thus we lose 60 grammes of power by employing two electro-magnets instead of one, although the two are put in direct communication with the poles of the battery. We might think, at first sight, that there was some fault in the electric communications, but our calculations prove it to be otherwise.

In applying the formulas of Ohm to the two cases we find :

1st. That in the case of a single electro-magnet, the value of the intensity of the current, the E. M. F. of the Daniell battery being taken as 5973, is 4.95.

2nd. That in the case in which the two electromagnets are on two circuits, this intensity is represented by 2.79.

The combined forces of these electro-magnets then, arranged in these two ways, will be as the squares of the two quantities 4.95 and 2.79, and if

we admit that the force of the electro-magnet in the simple circuit would be 200 grammes, the following formula will give the force of the two others:

so the joint force of the two electro-magnets will be 128 grammes instead of 200, as produced by a single one. This is a still more marked difference than that shown by experiment. This effect is in consequence of the arrangement of the battery, which is already out of proportion to the resistance of the exterior circuit, as its resistance is nearly four times as great as the latter, being still less in proportion to the circuit composed of two branches. Indeed, the n E formula which represents in this case 2np+H'

the intensity of the current on each branch, and which, in the case of a battery arranged in

α

series, gives for conditions of maximum 2 p = H

or

α

H

P = 2'

b

and this shows that in this case the re

sistance of the battery should be half that of each circuit. Consequently, the number a of the elements in tension should be obtained by the formula 'n H

✓

2p'

which in the present case gives

As we cannot divide a cell into fractions, and as the couplings require numbers which are even divisors

of n, the voltaic combination which will best answer to these conditions of maximum will be that in which the elements, separate and combined, will come the nearest to a, b, and n, and we shall see that, in the present case, it is that which contains three elements in tension, each composed of three elements in quantity, which will be the most advantageous. Now, under these conditions the attractive force of the single electro-magnet would be 267 grammes, and that of the two combined 316 grammes. Thus we gain by employing two electro-magnets.

We see by this how important it is that the construction of electro-magnets and the arrangement of the battery in electric applications should be preceded by calculation, and how much would be gained if the theory we are now propounding were well understood by constructors.

CHAPTER VI.

EXPERIMENTAL VERIFICATION OF THE LAWS OF ELECTRO-MAGNETS.

As I have already said, the formulas I have given have all been verified by experiment, except that generally adopted, which presents great difficulties in its proof, as we shall presently see. This experimental verification has seemed to me the more important, as originally many scientists did not accept these formulas without discussion, and besides, mathematical deductions are not always sufficient to convince. I have, therefore, undertaken several series of experiments to prove the truth of my deductions, and the following is the result:

1st. The experimental demonstration of the principle, establishing the fact that the resistance of the electro-magnet should equal that of the exterior circuit, is rather delicate, by reason of the difficulty we meet with in obtaining in all the wires of commerce of different diameters conductibility exactly proportional to the squares of those diameters. The experiments I have made have been rather contradictory, and I may say contrary to theoretic deductions, which made me hesitate, in my early investigations, in admitting them, although they were