168 
MR. C. V. BOYS OY THE RAniO-MICROMETER. 
From these it is seen that that length and size of wire is best of which the weight 
as much exceeds the dead weight as the resistance exceeds the dead resistance. In 
the same way, the efficacy of the best circuit may be showm to be 
best E,,,/ 
_ 1 _ 
8 y (CAV?fr) + pu'v 
( 6 ). 
If, finally, the breadth be made a vnrialfie, the following equations wall give the 
best conditions ;—■ 
As before. 
best o = 
and this is true whatever length, number of turns, wddth, or even shape the circuit 
may have. When h is greater than 1, 
for the best length, 2/ = 26 — 1 
for the best breadth, 26 = 2/ — 1 
. L 
ffi ^ + \f ~7~ ! 
\ u V 
L L /CW 
+ "+ V 
where e is the excess used for soldering. 
Therefore, for any breadth the length must exceed the breadth by as much as the 
breadth should exceed the length when that is given. In other wmrds, the>circuit is 
improved by adding to it ad infinitum, so that a square of infinite size is the best 
rectangle. If it should happen that ^ (CW/ffi-y) + e < 1, then the circuit in the same 
way would be made worse by increasing the dimensions. As a matter of fact, in the 
particular case, taking e as 0, this quantity is equal to 8'24. 
If, on the other hand, 6 is less than 1, then the quantity 26 — 1 + c in the twm 
equations above must be replaced by 6+1+6, a quantity necessarily positive ; 
hence, whether 6 is less than or greater than 1, the circuit cannot be too large. 
If the same process that has been followed in finding the best conditions wuth respect 
to weight be employed to find them with respect to moment of inertia, a difficultv 
arises in consequence of the fact that the upper end or cross wire of the circuit has a 
resistance which depends upon its length simply, while it has a moment of inertia which 
is only one-third of what it would have if it were placed alongside of one of the side 
wires of the circuit. Thus, while the expression for r remains as before, viz., 
(2/ + 1 + e) ^ a, that for the moment of inertia of the wire will be {(6? + 3c 
+ l)/12} u'a. If with these values the attempt is made to find the best values for a 
and I wdth respect to moment of inertia, a complicated cubic equation results and 
symmetrical expressions can no more be obtained. If, however, the two coefficients 
in parentheses had the same value, there would he no difficulty. 
While trying to find a remedy for this difficulty, I noticed that a, wure of uniform 
section is not the best form of conductor when the moment of inertia is taken into 
