174 
MR. C. V. BOYS ON THE RADIO-MICROMETER. 
On diiferentiating the expression for S with respect to a (the sectional area of the 
wire) and equating to 0, the maximum sensibility will he found when a has the same 
value ^/(Krj/wC), which gave the maximum efficacy ; on the other hand, the differential 
coefficient with respect to I (the length of the rectangle of copper) is negative for all 
positive values of I, that is, however short I may be, provided that the magnetic field 
may be made strong enough to keep the motion dead beat, a still greater sensibility 
Avill be given by a shorter circuit. If the strongest magnetic field conveniently 
available is not sufficient to make the circuit having maximum efficacy dead beat, 
then that circuit is still the best. If, however, as will be the case ^vith an ordinary 
magnet, the field is more than strong enough, then the length of rectangle must be 
reduced until the motion is deat beat. Taking the particular arrangement already 
referred to, in which the circuit is 1 cm. wide and is made of wire having the best 
sectional area ('001017 sq. cm.), the following are the values of the dead heat magnetic 
field for various lengths of circuit :— 
Best. 
Length . . '2 '4 '6 '8 I'O To 2'0 2'336‘6 
H' . . . 1722 929 664 532 455 347 294 27r8 
It is interesting here, in the expression for H' (12), to substitute those values of 
K and G which belong to the circuit of greatest efficac 3 ^ If this is done, there finally 
results the equation 
= 8 /y/-x/wt; ........ (12a), 
where H'' is the dead beat magnetic field for the circuit of greatest efficacy. It thus 
appears that, no matter what the resistance of the antimony-bismuth bars, or what 
the momentr of inertia of these bars, the mirror, and stem may be, provided that the 
circuit is so formed as to produce the greatest ultimate sensibility in aii}^ given field, 
the motion will he dead beat for one particular strength which simply depends upon 
the specific gravity and the specific resistance of the material with which the circuit 
is completed. 
The sensibility obtained by such a combination is 
1 
v/CK + 2 ? 
(13a). 
