[Footnote: Read at a meeting of the Physical Society, Feb. 26.]
By C.V. BOYS.
All the integrating machines hitherto made, of which I can find any
record, may be classed under two heads, one of which, Ainslee's machine,
is the sole representative, depending on the revolution of a disk which
partly rolls and partly slides on the paper, and the other comprising
all the remaining machines depending on the varying diameters of the
parts of a rolling system. Now, none of these machines do their work
by the method of the mathematician, but in their own way. My machine,
however, is an exact mechanical translation of the mathematical method
of integrating y dx, and thus forms a third type of instrument.
The mathematical rule may be described in words as follows: Required the
area between a curve, the axis of x and two ordinates; it is necessary
to draw a new curve, such that its steepness, as measured by the tangent
of the inclination, may be proportional to the ordinate of the given
curve for the same value of x, then the _ascent_ made by the new curve
in passing from one ordinate to the other is a measure of the area
required.
The figure shows a plan and side elevation of a model of the instrument,
made merely to test the idea, and the arrangement of the details is not
altogether convenient. The frame-work is a kind of T square, carrying a
fixed center, B, which moves along the axis of x of the given curve, a
rod passing always through B carries a pointer, A, which is constrained
to move in the vertical line, ee, of the T square, A then may be made
to follow any given curve. The distance of B from the edge, ee, is
constant; call it K, therefore, the inclination of the rod, AB, is such
that its tangent is equal to the ordinate of the given curve divided
by K; that is, the tangent of the inclination is proportional to the
ordinate; therefore, as the instrument is moved over the paper, AB has
always the inclination of the desired curve.
The part of the instrument that draws the curve is a three-wheeled cart
of lead, whose front wheel, F, is mounted, not as a caster, but like the
steering wheel of a bicycle. When such a cart is moved, the front wheel,
F, can only move in the direction of its own plane, whatever be the
position of the cart; if, therefore, the cart is so moved that F is in
the line, ee, and at the same time has its plane parallel to the rod,
AB, then F must necessarily describe the required curve, and if it is
made to pass over a sheet of black tracing paper, the required curve
will be _drawn_. The upper end of the T square is raised above the
paper, and forms a bridge, under which the cart travels. There is a
longitudinal slot in this bridge in which lies a horizontal wheel,
carried by that part of the cart corresponding to the head of a bicycle.
By this means the horizontal motion communicated to the front wheel of
the cart by the bridge, is equal to that of the pointer, A; at the same
time the cart is free to move vertically.
The mechanism employed to keep the plane of the front wheel of the cart
parallel to AB is made clear by the figure. Three equal wheels at the
ends of two jointed arms are connected by an open band, as shown. Now,
in an arrangement of this kind, however the arms or the wheels are
turned, lines on the wheels, if ever parallel, will always be so. If,
therefore, the wheel at one end is so supported that its rotation is
equal to that of AB, while the wheel at the other end is carried by the
fork which supports F, then the plane of F, if ever parallel to AB, will
always be so. Therefore, when A is made to trace any given curve, F will
draw a curve whose ascent is (1/K) f y dx, and this, multiplied by K, is
the area required.
[Illustration: AN INTEGRATING MACHINE.]
Not only does the machine integrate y dx, but if the plane of the front
wheel of the cart is set at right angles instead of parallel to AB, then
the cart finds the integral of dx / y, and thus solves problems, such,
for instance, as the time occupied by a body in moving along a path when
the law of the velocity is known.
Some modifications of the machine already described will enable it to
integrate squares, cubes, or products of functions, or the reciprocals
of any of these.
Of the various curves exhibited which have been drawn by the machine,
the following are of special physical interest.
Given the inclined straight line y = cx, the machine draws the parabola
y = cx² / 2. This is the path of a projectile, as the space fallen is as
the area of the triangle between the inclined line, the axis of x, and
the traveling ordinate.
Given the curve representing attraction y = 1 / x² the machine draws the
hyperbola y = 1 / x the curve representing potential, as the work done
in bringing a unit from an infinite distance to a point is measured
by the area between the curve of attraction, the axis of x, and the
ordinate at that point.
Given the logarithmic curve y = e^x, the machine draws an identical
curve. The vertical distance between these two curves, therefore,
is constant; if, then, the head of the cart and the pointer, A, are
connected by a link, this is the only curve they can draw. This motion
is very interesting, for the cart pulls the pointer and the pointer
directs the cart, and between they calculate a table of Naperian
logarithms.
Given a wave-line, the machine draws another wave-line a quarter of
a wave-length behind the first in point of time. If the first line
represents the varying strengths of an induced electrical current,
the second shows the nature of the primary that would produce such a
current.
Given any closed curve, the machine will find its area. It thus answers
the same purpose as Ainslee's polar planimeter, and though not so handy,
is free from the defect due to the sliding of the integrating wheel on
the paper.
The rules connected with maxima and minima and points of inflexion are
illustrated by the machine, for the cart cannot be made to describe a
maximum or a minimum unless the pointer, A, _crosses_ the axis of x, or
a point of inflexion unless A passes a maximum or minimum.
* * * * *
UPON A MODIFICATION OF WHEATSTONE'S MICROPHONE AND ITS APPLICABILITY TO
RADIOPHONIC RESEARCHES.
[Footnote: A paper read before the Philosophical Society of Washington.
D. C., June 11, 1881.]
By ALEXANDER GRAHAM BELL.
In August, 1880, I directed attention to the fact that thin disks or
diaphragms of various materials become sonorous when exposed to the
action of an intermittent beam of sunlight, and I stated my belief that
the sounds were due to molecular disturbances produced in the substance
composing the diaphragm.[1] Shortly afterwards Lord Raleigh undertook
a mathematical investigation of the subject and came to the conclusion
that the audible effects were caused by the bending of the plates
under unequal heating.[2] This explanation has recently been called in
question by Mr. Preece,[3] who has expressed the opinion that
although vibrations may be produced in the disks by the action of the
intermittent beam, such vibrations are not the cause of the sonorous
effects observed. According to him the aerial disturbances that produce
the sound arise spontaneously in the air itself by sudden expansion due
to heat communicated from the diaphragm--every increase of heat giving
rise to a fresh pulse of air. Mr. Preece was led to discard the
theoretical explanation of Lord Raleigh on account of the failure of
experiments undertaken to test the theory.
[Footnote 1: Amer. Asso. for Advancement of Science, August 27, 1880.]
[Footnote 2: _Nature_, vol. xxiii., p. 274.]
[Footnote 3: Roy. Soc., Mar. 10, 1881.]
[Illustration: Fig. 1. A B, Carbon Supports. C, Diaphragm.]
He was thus forced, by the supposed insufficiency of the explanation, to
seek in some other direction the cause of the phenomenon observed, and
as a consequence he adopted the ingenious hypothesis alluded to above.
But the experiments which had proved unsuccessful in the hands of Mr.
Preece were perfectly successful when repeated in America under better
conditions of experiment, and the supposed necessity for another
hypothesis at once vanished. I have shown in a recent paper read before
the National Academy of Science,[1] that audible sounds result from the
expansion and contraction of the material exposed to the beam, and that
a real to-and-fro vibration of the diaphragm occurs capable of producing
sonorous effects. It has occurred to me that Mr. Preece's failure to
detect, with a delicate microphone, the sonorous vibrations that were
so easily observed in our experiments, might be explained upon the
supposition that he had employed the ordinary form of Hughes's
microphone shown in Fig. 1, and that the vibrating area was confined
to the central portion of the disk. Under such circumstances it might
easily happen that both the supports (a b) of the microphone might touch
portions of the diaphragm which were practically at rest. It would of
course be interesting to ascertain whether any such localization of the
vibration as that supposed really occurred, and I have great pleasure in
showing to you tonight the apparatus by means of which this point has
been investigated (see Fig. 2).
[Footnote 1: April 21, 1881.]
[Illustration: Fig. 2. A, Stiff wire. B, Diaphragm. C, Hearing tube. D,
Perforated handle.]
The instrument is a modification of the form of microphone devised in
1872 by the late Sir Charles Wheatstone, and it consists essentially of
a stiff wire, A, one end of which is rigidly attached to the center of
a metallic diaphragm, B. In Wheatstone's original arrangement the
diaphragm was placed directly against the ear, and the free extremity
of the wire was rested against some sounding body--like a watch. In the
present arrangement the diaphragm is clamped at the circumference like
a telephone diaphragm, and the sounds are conveyed to the ear through a
rubber hearing tube, c. The wire passes through the perforated handle,
D, and is exposed only at the extremity. When the point, A, was rested
against the center of a diaphragm upon which was focused an intermittent
beam of sunlight, a clear musical tone was perceived by applying the ear
to the hearing tube, c. The surface of the diaphragm was then explored
with the point of the microphone, and sounds were obtained in all parts
of the illuminated area and in the corresponding area on the other side
of the diaphragm. Outside of this area on both sides of the diaphragm
the sounds became weaker and weaker, until, at a certain distance from
the center, they could no longer be perceived.
At the point where we would naturally place the supports of a Hughes
microphone (see Fig. 1) no sound was observed. We were also unable to
detect any audible effects when thepoint of the microphone was rested
against the support to which the diaphragm was attached. The negative
results obtained in Europe by Mr. Preece may, therefore, be reconciled
with the positive results obtained in America by Mr. Tainter and myself.
A still more curious demonstration of localization of vibration occurred
in the case of a large metallic mass. An intermittent beam of sunlight
was focused upon a brass weight (1 kilogramme), and the surface of the
weight was then explored with the microphone shown in Fig. 2. A feeble
but distinct sound was heard upon touching the surface within the
illuminated area and for a short distance outside, but not in other
parts.
In this experiment, as in the case of the thin diaphragm, absolute
contact between the point of the microphone and the surface explored was
necessary in order to obtain audible effects. Now I do not mean to
deny that sound waves may be originated in the manner suggested by Mr.
Preece, but I think that our experiments have demonstrated that the kind
of action described by Lord Raleigh actually occurs, and that it is
sufficient to account for the audible effects observed.
* * * * *
A catalogue, containing brief notices of many important scientific
papers heretofore published in the SUPPLEMENT, may be had gratis at this
office.
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