ME. C. V. BOYS ON THE RADIO-MICROMETER. 
177 
The cii’ciiit may be made smaller by reducing either factor, the length I or the 
breadth h. As has been shown, it is indifferent whether one or other factor is altered, 
provided that h is small compared with 1. Let the product be made N times as 
small, so that hi becomes 6//N, then the first factor 2/6/ in the expression for H', 
which it is convenient to repeat here in another form, 
will become 2N/6/. But when hi is changed, both k, the total moment of inertia, 
and B, the total resistance, are changed also. In the circuit of greatest efficacy the 
resistance and moment of inertia of the hoop are each equal to the fixed or invariable 
resistance and moment of inertia; and thus, when hi becomes 6//N, one half of each 
K and II also becomes one-N‘" of what it was, the other half of each remaining 
unchanged, and thus the new expression for H' becomes —/c'R, that is 
^ (N-f-1) times what it was. Therefore, if it is desired to make the dead beat 
magnetic field M times that which has been found for the circuit of greatest efficacy, 
the product hi must be reduced until it is 2M — 1 times as small, for N = 2M — 1. 
By this process the actual sensibility is increased, and the amount of increase may 
be found as follows:—The sensibility of any combination varies directly as the 
magnetic field and as the product hi, and inversely as the total moment of inertia 
and the total resistance; of these four quantities it has just been shown, that when 
hi is divided by N, the dead beat magnetic field is multiplied by ^ (N fi- l), and at 
the same time R and k are each multiplied by (N + l)/2N ; therefore, the sensibility 
of the arrangement becomes 
N + 1 1 2N 2N , , . 2N 
“A’ ^ N ^ N + 1 ^ N +1 ’ N + 1 ’ 
or 2 — (1/M) times what it was, so that 
S becomes (2 - i) 
Since the dead beat magnetic field for the circuit of greatest efficacy is about 
272 units, N must be so chosen as to make H' four or five times as great. Assuming 
that H' is to be increased to four times its original value, or tliat M = 4, the sensi- 
hility will only become If times what it was. Even in the case of an infinite field, it 
cannot be more than double that due to a field of 272 units if the motion is only just 
dead beat. From this it appears that, as long as the circuit has dimensions which at 
all approximate to those which theoretically are best, the sensibility obtained by 
moving the pole pieces until the dead beat conditions or the desired logarithmic 
decrement are produced is practically the highest which is possible. 
MDCCCLXXXIX.—A. 2 A 
