628 LENZ ON ELECTRO-MAGNETISM. 



In tlie same manner we find 

 Pi = 



P^ = 



^" ~ « ,r (2 nq + n"- (6 + |3) ) + /« (b + ^) ' " ^ '^ 



If I ditterentiate this general expression of the power of the current 

 for n series of convolutions in regard to w, I obtain 



d'p„ _ ^ JO ^ a'ir('2nq + n"~(b-\-^)) + m(b+ (3)-aitn(2q + 2n(b + ^)) 

 dn ^ [a it{2nq + n~ {b+ ^)) + m (b + /3)]'^ 



If I put this expression = 0, we have after some reductions 

 m — UTf n- =: 0, 



consequently " ^^ A / ( — I* 



I take here the positive sign of the root, because n according to its 

 nature cannot be negative, and ?n, a, it, are all three positive. 



If we further develope -^^ and put in the expression foimd this 

 dn'^ 



value of M = A / ( — ) we obtain a negative magnitude ; conse- 

 quently this value of n represents a maxinmm of the current. 



From the discovered value of n for the maximum of the current, we 

 can infer 



1. That the maximum of tlie action of tiie magnet on our spirals, for 

 every thickness of the wire, is attained by the same number of series 

 of convolutions ; for a is independent of 6 + /3. 



2. That the longer the free ends of the spirals are, or the greater m 

 is, the greater is the number of the series of convolutions required in 

 order to attain the maximum of the action. 



3. That the longer the space a is on which the convolutions may be 

 wound round in one series, the less number of series of convolutions 

 are necessary to produce the greatest current. 



4-. That the maximum is independent of q, i. e. that it is quite in- 

 different for the number of series of convolutions necessary to the at- 

 tainment of the maximum, whether they are immediately wound round 

 the cylinder of iron, or round another cylinder which is placed on the 

 other one. 



If we put the above found value w = . / ( ^ j in the general ex- 

 pression of the power which is contained in the equation (D.), we 

 obtain after some reductions, as the expression for the maximum which 



