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TENTH ANNUAL REPORT OP 



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 resist- 

 ance 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 



r -\- r 2 r 



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 

 Fi?- 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 tliis 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 num- 

 ber 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, . . . ?i times as long, 

 by placing 2, 3, . . . n times as many elements in succession, then 

 we shall have in all, 4, 9, . . . n~ 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, ... % times greater. 

 With 4a, '9«, . . . . nr a cups we can, in the most favorable case, ob 



t.ain 2, 3, n times as great a 



that which can be produced witli a elements 



strength 



of current as that 



§ 24. The most suitable arrangement of the closing arc for obtaining 

 a maximum effect luith 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 cop- 

 per 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 con- 

 veniently filled by the coils ; in that of the electro-magnets the quan- 



