626 LENZ ON ELECTUO-MAGNETISM. 



In the same manner we find, if three series have not a more power- 

 ful action than two 



i. e. this happens when the length of the free ends is half as great as 

 half the sum of the differences between the length of the first series and 

 the lengths of the second and third series. 



Having thus proved that by increasing the series of convolutions we 

 never obtain a maximum of the electric current, and therefore that a 

 greater increase would only do harm, we proceed to the general consi- 

 deration of the subject. 



We therefore suppose the convolutions of a series of the bespun me- 

 tallic wire to lie thick on one another. Let the length of the space on 

 which the convolutions may be wound up be = a, the thickness of the 

 wire = ft ; let the thickness of the wire covered with silk surpass the 

 thickness of the uncovered wire by the excess /3, so that it be = 6 + /3, 

 the length of a convolution be = c, the lengths of the free ends of the 

 wire = m ; the number of convolutions then which can be wound in 



one series upon the armature is = and the length of the wire 



of these convolutions = . c, and the whole length which the elec- 



o + p 



tricity has to run through for one series of convolutions 



■=. ^- c + m. 



If we assume the resistance offered by a wire of the same substance, 

 whose length = 1, and whose thickness = 1, as unity, the resistance 

 for one series of convolutions becomes 



c + m 



b + ^ _ ac + ( b + ^)m 



1,1 - b^(b + fy • 



Further, let the electromotive power produced in one convolution, 

 which, according to the second and third of our laws above proved, 

 remains the same for every magnitude of the convolutions and for 

 every thickness of the wire, be called f; the electromotive power pro- 

 duced in a series of convolutions is therefore, according to the first of 

 the above laws, 



- a f 

 b + ^-^' 



and consequently the power of the electric current for a series of con- 

 volutions, or 



_ a b\f 



^^' ac + (b + ^)m 



We must now for our purpose express the length of a convolution or c 



