628 LENZ ON ELECTRO-MAGNETISM. 
In the same manner we find 
ss Sa buf 
Ps amvt(6qg+9(6+ 6))+m(b6 +4 B) 
ae 4abef 
Ps Ga(8q+16(6 + B))+m(b +B) 
Pr fee 2a (D.) 
~ am (ang+nm(b+B))+m(b4 8) 
If I differentiate this general expression of the power of the current 
for 2 series of convolutions in regard to 2, I obtain 
d's |, Bef an(2Qng-+n?(b+B))+m(b+P)—arn(2q+2n(b+B)) 
dn law(2ng+n*(6+ B))+m(b + B))? 
If I put this expression = 0, we have after some reductions 
m—antn? =0, 
consequently n= a/ ) 
anv 
I take here the positive sign of the root, because » according to its 
nature cannot be negative, and m, a, m, are all three positive. 
If we further develope GPs and put in the expression found this 
: m : : ; 
value of 2 = a/ ( *) we obtain a negative magnitude; conse- 
ang 
quently this value of 2 represents a maximum of the current. 
From the discovered value of m for the maximum of the current, we 
can infer 
1. That the maximum of the action of the magnet on our spirals, for 
every thickness of the wire, is attained by the same number of series 
of convolutions ; for 7 is independent of 6 + £. 
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 g, 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 x = a/ ( ~) in the general ex- 
ang 
pression of the power which is contained in the equation (D.), we 
obtain after some reductions, as the expression for the maximum which 
