362 TENTH ANNUAL REPORT OP 



we can doiilile the widtli of eacli element — we fihall then have 6 four- 

 fold elements. 



Making the pile three times shorter, three times as many single 

 elements can he united in one ; from 12 douhle elements we ohtain 4 

 of six-fold. In short, if the pile he 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 tiie battery had now been made only a times shorter, without in- 

 creasing 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 o? 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 I. This battery being closed by a 

 conducting circuit, whose resistance is also I, we have, according to 

 Ohm's law, the strength of the current — 



The pile being now made a times shorter, but the single elements a 



E 

 times wider_,the^lectro-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 



E 



Q' — ^ _ E 



~l_+ l~ l{a+l\ ^ (2) 



«- a) 



But the sum a -\- — h, 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 there- 

 fore proved. 



The application of this proposition may be shown by an example. 

 If, in magnetizing an electro-magnet, the current of 24 zinc and. car- 

 bon 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 luire, and therefore 

 the resistance of the connecting arc is 0.9 of that of a single ele- 



