LENZ ON ELECTRO-MAGNETISM. 625 



the length of the cylindrical ai'mature; therefore the further increase of 

 the number of convolutions can only be made by several series of con- 

 volutions placed one above another. Let the electromotive power of a 

 series of convolutions which the length of the armature occupies = (p ; 

 the length of the wire of all these convolutions, or, in this case, on ac- 

 count of the diameter of the wire being equal throughout, the resist- 

 ance it oiFers = a ; let the length of the necessarily free ends of the 

 wires together = /3, the power therefore of the current of this first se- 

 ries of convolutions is 



let Y be the piece of the second series of convolutions by which its 

 length, on account of its necessarily greater diameter, is greater than 

 the length a of the first series, the power of the current from these 

 two series is 



20 

 '^'-2a + 7 + ^' 

 and in the same manner 



3<p 



'^^ - 3 a + 7 + + /3' 

 where S designates the quantity by which the first series is surpassed in 

 length by the third. If now the second series of convolutions does not 

 add to the strength of the current, we put fx,, = /x^ j therefore 



(^ 2^ 



a + /3 ~2a -+ /3 -Hy' 

 whence we have g ^:; y 



i. e. as soon as the length of the free ends is only equal to the differ- 

 ence between the lengths of the second series of convolutions and those 

 of the first, the second series would then add nothing to the strength 

 of the current. In order to see what three series would do in this case, 

 let us put /3 = y in the expression for |x, and we obtain 

 _ 3<p 



^' Sa4-2/3+ 5' 

 S however is now greater than y or /3, we therefore put iJ = /3 + ^, 

 where /* expresses a positive magnitude ; we obtain by this 



3«p _ <}> 



f*3 



3a+3/3 + f. (^^^) + ii 



3 



This last expression for |X3 is evidently smaller than — ^TT^ conse- 

 quently three series of convolutions would only weaken the action of 

 one or two series (whicii actions have been here assumed as o(jual). 



