LENZ ON ELECTRO-MAGNETISM. 625 
the length of the cylindrical armature; 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 = 9; 
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 offers = a; let the length of the necessarily free ends of the 
wires together = 6, the power therefore of the current of this first se- 
ries of convolutions is 
SAD yi: 
a aa B 
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 
ihe orid ita F 
Ko — 2a aR y os B p) 
and in the same manner 
3 
~B8aty+i+P 
where # 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 ~, = (2, therefore 
BR ca I 
a+~Bo 2at+Bry 
Pat 
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 6 = y in the expression for 4, and we obtain 
alt or Siena 
Os 3a +2 B+? 
3 however is now greater than y or 8, we therefore put 9 = B + p, 
where pz expresses a positive magnitude ; we obtain by this 
ie ae 
Bet BE Eee e 
bs 
whence we have 
This last expression for us is evidently smaller than conse. 
9 
a+ 6 
quently three series of convolutions would only weaken the action of 
one or two series (which actions have been here assumed as equal). 
