MR. C. BOYS OX THE RADIO-MICROMETER. 
IG9 
account, i.e., of that part of the wire which crosses the axis. That this is so is evident 
in two ways. Since a wire of uniform section has the least resistance for a given 
length and weight, it cannot iiave the least resistance for a given length and moment 
of inertia, except when it is parallel to the axis of rotation. Or, considering the cross 
piece only, since the uniform section is that which gives the least resistance for its 
weight, it is clear that a very small change of form, such for instance as would occur if a 
thin skin were removed from the outer and were transferred to those portions nearer 
the axis of rotation, would produce no appreciable change in resistance, but it would 
reduce the moment of inertia ; therefore, there is an advantage in having tliose parts 
near the axis thicker than those more distant from it. If we suppose that the 
resistance of any element of the wire between the axis and the ends is inversely 
proportional to the cross section, and that the resistance of the whole piece is the 
sum of the resistances of the several elements, which is true as long as the rate of 
change of section is not so rapid as to make the stream lines notably inclined, then 
the best variation of section will be that in which no change will be made in the 
resistance -j- the moment of inertia if a thin skin be transferred from one part to 
another. Taking the axis as origin of rectangular coordinate, and calling the 
sectional area y, then for all values of x, Ijy yx^, i.e., Ijy'^x^ must be constant or y 
must vary inversely as x. It thus appears that, if the hoop is cut out of sheet copper 
instead of wire, that part which forms the upper bar should be bounded by two 
hyperbolas of such dimensions that the width at the ends of the cross bar is the same 
as that of the side pieces, which are uniform. If the metal is chosen of such a thiclc- 
ness, t, that the best section of the sides hereafter to be found = ^ X ’1 mm., then 
the error due to the inclination of the stream lines already pointed out will not be 
appreciable, except within about \ mm. on either side of the axis. It will be well at 
once to point out that the .value of this supposed form for the cross bar does not at 
all lie in the fact that it is the best form, for nothing worth consideration would 
be gained by adopting it, but the reason for bringing it forward is this : the total 
resistance of either half of the cross bar will be, if the resistance of the same length 
of the wire be called 1, = fxdx = ^; the moment of inertia of either half will be, 
if that of the same length of the side be called 1, = dx = -g. That is, not only 
has the best form been found for the cross bar, but the coefficients for moment of 
inertia and for resistance have at the same time been made identical, and thus all the 
expressions found with respect to weight will equally apply with respect to moment 
of inertia. 
It may be objected, since, as already mentioned, the resistance of these parts 
close to the axis is greater than is supposed, on account of the inclination of the 
stream lines and the consequent concentration of the current in the more direct line, 
or again, since the weight of a piece of metal filling the space between two hyperbolas 
is infinite, that the solution for the difficulty thus put forward is not correct, and that 
any further calculations based upon this result will not be trustworthy. The answer 
MDCCCLXXXIX.-A. 
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