we can double the width of each element-we shall then have 6 fourfold elements.

Making the pile three times shorter, three times as many single elements can be united in one; from 12 double elements we obtain 4 of six-fold. In short, if the pile be made a times shorter, we can unite a times as many single elements in one.

If the number of elements combined, one after another, to form a pile, is a times less, the electro-motive force thus becomes a times less; if the battery had now been made only a times shorter, without increasing its width, the resistance would have been a times less; but if each element of those in a pile consists of a times as many single elements as before, the resistance becomes a2 times less than before.

Thus the resistance of 6 quadruple elements (combination No. 4) is 4 times less than for 12 double elements, (combination No. 2;) for 4 six-fold elements (combination No. 5) 9 times less than for 12 double, &c.

From this exposition the proof in question is easily derived. For any combination of a number of elements, let the electro-motive force be E, and the battery resistance 7. This battery being closed by a conducting circuit, whose resistance is also l, we have, according to Ohm's law, the strength of the current—

S=

E

E
1+1=21⋅

(1)

The pile being now made a times shorter, but the single elements a E times wider, the electro-motive force will be a times less, or; but the resistance of the battery will be and the force of the current, for

the same connecting arc, will be

S'

a

E

-

a

E

1 + 1 = 1 ( a + 1)

42

a

(2)

But the sum a + is, under all circumstances, greater than 2*, which, in an integral or fractional quantity we may substitute for a; thus the value of the fraction (2) is, under all circumstances, less than that of (1.) Since (1) denotes the value of the strength of the current for cases in which the resistance in the electrometer is equal to the resistance of the closing arc, and the fraction (2) the value of the strength of current for cases in which the number of single elements is combined in any other manner, the proposition in question is therefore proved.

The application of this proposition may be shown by an example. If, in magnetizing an electro-magnet, the current of 24 zinc and carbon elements be used, the resistance of one element, with weak acid, is 15.05. But resistance of the coils of the electro-magnet has been found equal to that of 13.54 metres of normal wire, and therefore the resistance of the connecting arc is 0.9 of that of a single ele

ment. A glance at the arrangements (Figs. 14 and 15) shows us that we must select the fifth combination as the most suitable; because its resistance, 0.65, is nearer to that of the closing arc, than that of the other combinations. Make, for sake of brevity, the electro-motive force of the element equal 1, and the resistance also 1, then, if we apply successively all of the eight combinations to the electro-magnet above mentioned, the following values will be obtained for the strength of the current:

It is observed here that with the combination 5 the coils of the electro-magnet remaining unchanged, the magnetism of the soft iron will be greater than with any of the other combinations. Combination 4 approaches 5 very closely in its effects; thus the exact maximum should be looked for between 4 and 5. In fact the combination represented in fig. 16 gives the strength of the current 2.56.

Fig. 16.

By charging the same elements with strong acid, the resistance of the element will be 5.85; the resistance of the closing arc will be 2.3 times as great as that of one element, and for this case the third combination (eight three-fold elements) will be the most suitable.

The best combination for a given apparatus to decompose water will be further considered hereafter.

If a given number of elements be so conbined that they will yield in a given circuit a maximum strength of current, an increase of the number of elements will increase the strength of the current in the most favorable cases only

in proportion to the square root of the number of elements; then 4, 9, or 16 times as many elements must be used to obtain 2, 3, or 4 fold effects.

We shall endeavor to prove this, in a special case. Let the resistance of the closing arc be r, equal to the resistance of one element, the electro-motive force of which is denoted by E, then the strength of the current is

Now let us double the force of the current by increasing the number of elements. To obtain a maximum effect from the new combination, the resistance in the battery must continue as great as the resistance of the closing arc; therefore, the resistance of the new combination must not be greater than that of a single element; hence, we shall obtain double the force of the current if, with unchanged resistance, we

Fig. 17.

double the electro-motive force. This is done by placing one element after another; but we must take 2 double elements, if their resistance is to be as great as that of a single element; hence, the combination of Fig. 16 will give twice as great a force of current, and Fig. 17 three times as great, as a single element.

.

To consider this matter in a more general way, let a number of cups a be so combined, that the resistance of the battery is equal to that of the conducting circuit, so that we attain the maximum effect which the number a of cups can produce in the given closing arc. Place 2, 3, . . . n times as many cups together, so that each element of the battery may have 2, 3, .n as great a surface; but if the battery is made at the same time 2, 3, . . . n times as long, by placing 2, 3,... n times as many elements in succession, then we shall have in all, 4, 9, . . . n2 times as many cups in use. The resistance of the battery by this arrangement remains unchanged, and therefore the strength of the current increases in the same ratio as the electro-motive force, namely, in the ratio of the number of successive elements; it has thus become 2, 3, . . . n times greater. With 4a, 9a,. . . . n2 a cups we can, in the most favorable case, obtain 2, 3,. n times as great a strength of current as that that which can be produced with a elements.

§ 24. The most suitable arrangement of the closing arc for obtaining a maximum effect with a given electro-motor.-In some cases the electro-motor is given, and the question is, how the coils of wire must be selected to obtain a maximum effect; from the same quantity of copper are many coils of a thin and long wire to be made, or fewer coils with short and thick wires? In the case of multipliers, the quantity of copper wire to be used is limited by the space which can be conveniently filled by the coils; in that of the electro-magnets the quan

tity of copper wire is limited by the amount of money to be expended in its construction.

Suppose the resistance of a copper wire of a given length and thickness, making n coils, to be equal to 1, or the resistance of the electromotor; then the force of the current is

and this acting in n coils on the magnetic needles in soft iron, we can. represent its effect by

If we make the wire m times as long, the mass remaining the same, its section will be m times less, and then the resistance m2 times greater; hence the force of the current is now

but of this length of wire, m times as many coils can be made as before; thus, the magnetic effect is now

But the value of M, as just proved, is always greater than the value of M'. Hence with a given mass of wire, a maximum of magnetic effect is obtained by giving to the wire such a thickness and length that the resistance in the coils is equal to that of the elements.

For instance, if we have eight pounds of copper wire for constructing an electro-magnet, to be excited by one of Daniell's elements, described in section 9, how thick must the wire be made?

The resistance of this element is equal to the resistance of 11.1 metres of the normal wire. The normal wire has a section of 0.785 of a square millimetre, or 0.00785 of a square centimetre; thus, a length of 11.1 metres or 1,110 centimetres has a cubic contents of 8.71 cubic centimetres. The specific weight of the copper to be drawn to wire is 8.88; hence the weight of the normal wire, which has the same resistance to conduction as the element, is 8.71 x 8.8877.34 grammes.

But the mass of wire which we have at our disposal does not weigh 77.34 grammes, but eight pounds, or 4,000 grammes; so that we have 19951.7 times as great a mass as that of the normal wire which fulfils the condition.

4000

If, instead of a wire of given diameter and length, one of three times the diameter be taken, its section is 3 x 39 times greater, and a nine-fold length must be given to it, that it may retain its resistance to conduction unchanged; the volume of the wire is now 81=34 times as great as it was before. A wire n times as thick must have a length n as great, and consequently n' greater mass, if its resistance is to remain unchanged.

Hence, with a mass p times as great, the wire muзt have a length √p times as great, and a diameter √p times; the resistance remaining invariable."

The mass of copper to be disposed of is 51.7 times as great as that of a normal wire which offers the same resistance as the elements; hence, we must make of this mass, a wire which is √51.7=7.18 times as long, and √ 51.7 2.68 times as thick as the normal wire, 11.1 metres long. Thus, if the eight pounds of copper wire is to oppose the same resistance as the Daniell's element, it must be 2.68 millimetres thick, thus requiring a length of 7.18 x 11.1 79.7 metres. If the electro-magnet is to be arranged for a Stöhrer's element, whose essential resistance is equal to that of 6.2 metres of the normal wire, for the same reason, the eight pounds of copper must be a wire 3.1 millimetres thick; which requires a length of 60 metres.

Using the electro-magnet constructed for Daniell's battery, with this battery, the strength of the current is

The wire being placed in n coils about the iron, the magnetic effect may be denoted by

Had the wire been twice as long, and consequently one-half in section, its resistance would have been four times as great, or 44.4, and the strength of the current

but this is passed around the iron in 2 n coils, and the magnetic effect is now

If a wire half as long but double in section had been used, the magnetic effect would have been

Thus it is seen that the values of M' and M" are less than that of M. According to these principles, we can also determine how, with a given thermo-electric battery, a multiplier of the greatest possible sensibility may be constructed-a question which was solved theoretically long since, but until now the solution has not had a form susceptible of practical application. On this account we shall give this subject some further consideration.

For instance, our physical cabinet possesses a thermo-electric pile with the galvanometer belonging to it. I found the Resistance of the thermo-pile......... "wire of multiplier......

18.34 met. of normal wire. 1.75"