Referring again to fig. 1, we there see the base of the curve _A B_
divided into the equal parts _a b_, _b c_, and _c d_; and observing the
passage of the generating circle _D_, from the origin of the curve at
_d_, to the first division _c_ on the base, we shall find no more than
the small portion _d e_, of the curve developed, whereas a second equal
step of the generating circle _c b_, will extend the curve forward from
_e_ to _f_, a greater distance than the former; while a third equal step
_a b_, will extend the curve from _f_ to _g_, a distance greater than
the last; and the successive increments of the curve will be still
greater, as it approaches its summit; yet all these parts correspond to
equal advances of the wheel, namely, to the equal parts _a b_, _b c_ and
_c d_ of the base, and to equal ones of rotation of the generating
circle. Surely then the parts _s g_, _g f_, of the epicycloidal tooth
will be _worn out_ sooner than those _f e_, _e d_, which are rubbed with
so much less velocity than the other, even though the _pressure_ were
the same. But the pressure is not the same. For, the line _a g_ is the
direction in which the pressure of the curve acts at the point _g_, and
the line _p q_, is the length of the lever-arm on which that pressure
acts, to turn the generating circle on its axis (now supposed to be
fixt;) but, as the turning force or rotatory effort of the wheels, is by
hypothesis uniform, the pressure at _g_ must be inversely as _p q_; that
is, inversely as the cosine of half the angle of rotation of the
generating circle; hence it would be infinite at _s_, the summit of the
curve, when this circle has made a semi-revolution.
Thus it appears that independently of the effects of percussion, the
_end_ of an epicycloidal tooth must _wear out_ sooner than any part
nearer its base, (and if so, much more it may be supposed of a tooth of
another form;) and that when its form is thus changed, the advantage it
gave must cease, since nothing in the working of the wheel can
afterwards restore the form, or remedy the growing evil.
Having now shewn one great defect in the common system of wheels, I
shall proceed to develope the principles of the new system, which may be
understood through the medium of the three following propositions.
1. The action of a wheel of the new kind on another with which it works
or _geers_ is the same at every moment of its revolution, so that the
least possible motion of the circumference of one, generates an exactly
equal and similar motion in that of the other.
2. There are but two points, one in each wheel, that necessarily touch
each other at the same time, and their contact will always take place
indefinitely near the plane that passes through the two axes of the
wheels, if the diameters of the latter, at the useful or pressing points
are in the exact ratio of their number of teeth respectively; in which
case there will be no sensible friction between the points in contact.
3. In consequence of the properties above-mentioned, the epicycloidal or
any other form of the teeth, is no longer indispensable; but many
different forms may be used, without disturbing the principle of equable
motion.
With regard to the demonstration of the first proposition, I must
premise an observation of M. Camus on this subject, in his Mechanics,
3d. part, page 306, viz. “if all wheels could have teeth infinitely
fine, their _geering_, which might then be considered as a simple
contact, would have the property required, [that of acting uniformly]
since we have seen that a wheel and a pinion have the same _tangential_
force, when the motion of one is communicated to the other, by an
infinitely small penetration of the particles of their respective
circumferences.”
Now suppose that on the cylindrical surface of a spur-wheel _B c_, (fig.
3) we cut oblique or rather _screw-formed teeth_, of which two are
shewn at _a c_, _b d_, so inclined to the plane of the wheel, as that
the end _c_ of the tooth _a c_ may not pass the plane of the axes _A B
c_, until the end _b_ of the other tooth _b d_ has arrived at it, this
wheel will virtually be divided into an infinite number of teeth, or at
least into a number greater than that of the particles of matter,
contained in a circular line of the wheel’s circumference. For suppose
the surface of a similar, but longer cylinder, stripped from it and
stretched on the plane _A B C E_ (fig. 4) where the former oblique line
will become the hypothenuse _B C_, of the right angled triangle _C A B_,
and will represent _all_ the teeth of the given wheel, according to the
sketch _E G_ at the bottom of the diagram. Here the lines _A B_ and _C
E_, are equal to the circumference of the base of the cylinder, and _A
C_ and _B E_ to its length; and if between _A_ and _B_, there exist a
number, _m_, of particles of matter, and between _A_ and _C_ a number,
_n_, the whole surfaced _A B C E_ will contain _m n_ particles, or the
product of _m_ and _n_; and the line _B C_, will contain a number =
√(_m_² + _n_²), from a well known theorem; whence it appears that the
line _B C_ is necessarily longer than _A B_, and hence contains more
particles of matter.[2]
[2] It need hardly be observed, that whatever is true of the whole
triangle C A B, (fig. 4) is true of every similar part of it, be it ever
so small: and in fact, when the hypothenuse B C, is folded again round
the cylinder, from which we have supposed it stripped, the acting part
will be very small indeed; but it will still act in the way here
described, and give tendencies to the wheel it acts on, and to its axis,
precisely proportionate to the quantities here mentioned.
It is besides evident, that the difference between the lines _B C_ and
_A B_, depends on the angle _A C B_; in the choice of which, there is a
considerable latitude. For general use however, I have chosen an angle
of obliquity of 15°, which I shall now assume as the basis of the
following calculations. The tangent of 15°, per tables, is in round
numbers 268 to radius 1000; and the object now is to find the number of
particles in the oblique line _B C_, when the line _A B_, contains any
other number, _t_.
By geometry, _B C(x)_ = √(_r_² + _t_²) = √(1000² + 268²) = 1035 nearly;
and this last number is to 268, as the number of particles in the
oblique line _B C_ is to the number contained in the circumference _A
B_, of the base of the cylinder. Hence it appears, that a wheel cut into
teeth of this form, contains (virtually) about four times as many teeth,
as a wheel of the same diameter, but indefinitely thin, would contain.
And the disproportion might be increased, by adopting a smaller angle.
Thus I apprehend it is proved, that the action of a wheel of this kind,
on another with which it geers, is perfectly uniform in respect of
swiftness; and hence the proof that it is likewise so, as to the force
communicated.
Before I proceed to the second proposition, I ought perhaps to
anticipate some objections that have been made to this system of
geering, and which may have already occurred to some gentlemen present.
For example, it has been supposed that the _friction_ of these teeth,
is augmented by their inclination to the plane of the wheel; but I dare
presume to have already proved, that it is this very obliquity, joined
to the total absence of motion in direction of the axes, that _destroys_
the friction, instead of _creating_ it. I acknowledge however, that the
_pressure_ on the points of contact, is greater than it would be on
teeth, parallel to the axes of the wheels, and I farther concede that
this pressure tends to displace the wheels in the direction of the axes,
(unless this tendency is destroyed by a tooth, with two opposite
inclinations.) But supposing this counteraction neglected, let us
ascertain the importance of these objections. First, with regard to the
increase of pressure on the point _D_ of the line _B C_, (representing
the oblique tooth in question,) relative to that which would be on the
line _B E_, (which represents a tooth of common geering:) let _A D_ be
drawn perpendicular to _B C_. If the point _D_ can slide freely on the
line _B C_, (and this is the most favourable supposition for the
objection,) its pressure will be exerted perpendicularly to this line;
and if the point _A_, moves from _A_ to _B_, the point _D_, leaving at
the same moment the point _A_, and moving in direction _A D_, will only
arrive at _D_ in the same time, its motion having been slower than that
of _A_, in the proportion of _A B_ to _A D_; whence by the principle of
virtual velocities, its pressure on _B C_ is to that on _A C_, as the
said lines _A B_ to _D A_.
To convert these pressures into numbers, according to the above data; we
have _A C_ = 1000, _A B_ = 268, _B C_ = 1035; then from the similar
triangles _B A C_, _B D A_, it will be _B C_ : _A C_ ∷ _A B_ : _A D_ =
268000/1035 = 259 nearly. Therefore the pressure on _B C_, is to that on
_A C_, as 268 to 259, or as 1035 : 1000.
To find what part of the force tends to drive the point _B_, in the
direction _B E_, (for this is what impels the wheels, in the direction
of their axes,) we may consider the triangle _B A C_ as an inclined
plane, of which _B C_ is the length, and _A B_ the height; and the total
pressure on _C B_, which may be represented by _C B_, (1035) may be
resolved into two others, namely, _A B_ and _A C_, which will represent
the pressures on those lines respectively, (268 and 1000.) Hence the
pressure on _B C_, is augmented only in the ratio of 1035 to 1000, or
about 1/29 part by the obliquity; and the tendency of the wheels to move
in the direction of their axes, (when this angle is used,) is the
268/1000 of the original stress, that is, rather more than one quarter.
But since the longitudinal motion of an axis can be prevented by a point
almost invisible applied to its centre, it follows that the effect of
this tendency can be annulled, without any sensible loss of the active
power. It may be added, that in vertical axes, those circumstances lose
all their importance, since whatever force tends to _depress_ the one
and increase its friction, tends equally to _elevate_ the other, and
relieve its step of its load; a case that would be made eminently
useful, by throwing a larger portion of pressure on the _slow-moving_
axes, and taking it off from the more rapid ones.
We now proceed to the second proposition. The truth of the assertions,
contained in this proposition, must, I should suppose, be evident, from
the consideration of two circles touching each other, and at the point
of contact, coinciding with their common tangent at that point. Let _A_
and _B_ be two circles, tangent to each other, (fig. 3) in _e_. _A C_ is
the line joining the centres, and _D F_ the common tangent of the
circles at _e_; which is at right angles with _A C_; and so are the
circumferences of the two circles at the point _e_. For the circles and
tangent coincide for the moment. Hence then I conclude, 1st that a
motion (evanescently small) of the point common to the three lines, can
take place without quitting the tangent _D F_: and 2d. that if there is
an infinite number of teeth in these circles, those which are found in
the line of the centres, will _geer_ together in preference to those
which are out of it, since the latter have the common tangent, and an
interval of space between them.
The truth of this proposition (or an indefinite approximation to truth,)
may be deduced from the supposition that the two circles do _actually_
penetrate each other. To this end let _A B_ _a b_, in fig. 5, be two
equal circles, placed parallel to each other in two contiguous planes,
so as for one to hide the other, in the indefinitely small curvilinear
space _d f e g_. I say that if the arc _d g_ is indefinitely small, the
rotation of the two circles will occasion no more friction between the
touching surfaces, _g e f_ and _f d g_, than there would be between the
two circles placed in the same plane, and touching at the point n the
same common tangent.
For draw the lines _D E_, _f d_, _d g_, _g f_, _g e_ and _g D_; and
adverting to the known equation of the circle, let _d n_ = _x_, _g n_ =
_y_ and _D g_ = _a_, the absciss, ordinate and radius of the circle; we
have 2 _a x_ - _x_² = _y_². From this equation we obtain _a_ = (_y_² +
_x_²)/2_x_, the denominator of this fraction (2_x_) being the width, _d
e_, of the touching surfaces _f d g_, and _f e g_ of the two circles.
But the numerator (_y_² + _x_²) is equal to the square of the chord _g
d_ of the angle _E D g_, which chord I shall call _z_; then we have _a_
= _x_²/2_x_ from which equation we derive this proportion, _a_ : _z_ ∷
_z_ : 2_x_ = _z_²/_a_. But in very small angles, the sines are taken for
the arcs without sensible error; and with greater reason may the chords;
if then we suppose the arc _d g_, or the chord _z_, indefinitely small,
we shall find the line _d e_ = 2_x_ = _z_²/_a_, indefinitely smaller;
that is, of an order of infinitessimals one degree lower; for it is well
known that the square of evanescent quantities are indefinitely smaller
than the quantities themselves. And to apply this, if the chord _z_
represent the circular distance of two particles of matter found in the
screw-formed tooth _a c_, of the wheel _B c_, fig. 3, (referred to the
circle _a b_, fig. 5), that distance _z_ will be a mean proportional
between the radius _D g_ of such wheel, and the double versed sine of
this inconceivably small angle.[3]
[3] I ought perhaps to have introduced this reasoning on the 5th. figure
by observing, that every projection of every part of a screw, on a plane
at right angles with the axis of such screw, is a circle; and that
therefore the chord _z_, or the line _g d_, is the true projection of a
proportionate part of any line, _B C_, fig. 4, when wrapped round a
cylinder of equal diameter with the circle _a b_, fig. 5.
I am aware that some mathematicians maintain, that the smallest portion
of a curve cannot strictly coincide with a right line; a doctrine which
I am not going to impugn. But however this may be, it appears certain
that there is no such mathematical curve exhibited in the material
world; but only polygons of a greater or less number of sides, according
to the density of the various substances, that fall under our
observation. I shall therefore proceed to apply the foregoing theory,
not indeed to the ultimate particles of matter, (because I do not know
their dimensions,) but to those real particles which have been actually
measured. Thus, experimental philosophy shews, that a cube of gold of
1/2 inch side, may be drawn upon silver to a length of 1442623 feet, and
afterwards flattened to a breadth of 1/100 of an inch, the two sides of
which form a breadth of 1/50 of an inch: so that if we divide the above
length by 25, we shall have the length of a similar ribbon of metal of
1/2 an inch in breadth, namely, 57704 feet; which cut into lengths of
1/2 an inch, (or multiplied by 24, the half inches in a foot) give
1384896 such squares, which must constitute the number of laminæ of a
half inch cube of gold, or 2769792 for an inch thickness. Let us suppose
then a wheel of gold, of two feet in diameter, the friction of whose
teeth it is proposed to determine. We must first seek what number of
particles are contained in that part of the tooth or teeth, that are
found in one inch of the wheel’s circumference; this we have just seen
to be 2769792 thicknesses of the leaves, or diameters of the particles,
such as we are now contemplating.
We shall now have this proportion, (see fig. 4) 268 (_A B_) : 1035 (_B
C_) ∷ 2769792 (no. of particles in one inch of circumference of base) :
_x_ = 10696771 particles in that part of the line _B C_, which
corresponds with _that_ inch of the circumference. Thus each of the
latter particles measured in the direction _A B_, is equal to the
fraction 1/10696771ths of an inch. And if that fraction be taken for the
arc _g d_, (fig. 5) then to find the length of the line _d e_, (on which
the friction of _this_ and all other geering depends) we must use this
analogy; 12 inch (rad. of wheel) : 1/10696771 of an inch (chord _g d_) ∷
1/10696771 of an inch (_g d_) : _d e_, the line required =
1/1273050917917292 of an inch. This result is still beyond the truth, as
we do not know how much smaller the ultimate molecules of gold are.
To advert now to some of the practical effects of this system, I would
beg leave to present a _form_ of the teeth, the sole working of which
would be a sufficient demonstration of the truth of the foregoing
theory. _A_, _B_, (fig. 6) are two wheels of which the primitive circles
or pitch-lines touch each other at _o_. As all the homologous points of
any screw-formed tooth, are at the same distance from the centres of
their wheels, I am at liberty to give the teeth a rhomboidal form, _o t
i_; and if the angle _o_ exists all round both wheels, (of which I have
attempted graphically to give an idea at _D G_,) in this case, those
particles only which exist in the plane of the tangents _f h_, &c. and
infinitely near that plane passing at right angles to it through the
centres _A_ and _B_, will touch each other; and there, as we have
already proved, no sensible motion of the kind producing friction,
exists between the points in actual contact. I might add, as the figure
evidently indicates, that if any such motion did exist, the angles _o_
would quit each other, and the figure of such teeth become absurd in
practice; but on the other hand, if such teeth can exist and work
usefully (which I assert they can, nay that all teeth have in this
system a tendency to assume that form at the working points;) this
circumstance is of itself a practical evidence of the truth of the
foregoing theory, and of what I have said concerning it.
It must have been perceived that I have in some degree anticipated the
demonstration of my third proposition, namely, that the epicycloidal or
any other given form of the teeth, is not essential to this geering. It
appears that teeth formed as epicycloids, will become more convex by
working; since the base of the curve is the only point where they suffer
no diminution by friction; whilst those of every other form, that
likewise penetrate beyond the primitive circles of the wheels, will also
assume a figure of the same nature, by the rounding off of their points,
and the hollowing of the corresponding parts of the teeth they impel;
and that operation will continue till an angle similar to that at _o_,
but generally more obtuse, prevails around both wheels; when all
sensible change of figure or loss of matter will cease, as the wheels
now before you will evince.
On the right of the drawing, (fig. 6) the teeth of the wheel _B_ are
angular, (suppose square) and those of the wheel _C_ rounded off by any
curve _s_, _within_ an epicycloid. All that is necessary to remark in
this case is, that the teeth of the wheel _B_ must not extend _beyond_
its primitive circle, whilst the round parts of those of the wheel _C_,
do more or less extend beyond its primitive circle; whence it becomes
evident, that the contact of such teeth, (if infinite in number) can
_only_ take place in the plane of the common tangent at right angles to
_A B_; also that if these teeth are sufficiently hard to withstand
ordinary pressure, without indentation in these circumstances, there is
no perceptible reason for a sensible change of form; since this contact
only takes place where the two motions are alike, both in swiftness and
direction. A fact I am going to mention may outweigh this reasoning in
the minds of some, but cannot invalidate it. I caused two of these
wheels made of brass, to be turned with rapidity under a considerable
resistance for several weeks together, keeping them always anointed with
_oil_ and _emery_, one of the most destructive mixtures known for
rubbing metals; but after this severe trial, the teeth of the wheels,
_at their primitive circles_ were found as entire as before the
experiment. And why? Certainly for no other reason than that they worked
without sensible friction.
Hitherto nothing has been said of wheels in the conical form, usually
denominated _mitre and bevel geer_. But my models will prove, that they
are both comprehended in the system. The only condition of this unity of
principle is, that the axes of two wheels, instead of being _parallel_
to each other, be always found in _the same plane_. With this condition,
every property above-mentioned, extends to this class of wheels, which
my methods of executing also include, as indeed they do every possible
case of geering.
Being afraid of trespassing on the time of the society, I have
suppressed a part of this paper, perhaps already too long; but I hope I
may be indulged with a few remarks on the application of those wheels to
practical purposes. And first, as to what I have myself seen; these
wheels have been used in several important machines to which they have
given much swiftness, softness or precision of motion as the case
required. They have done more; they have given birth to machines of no
small importance, that could not have existed without them. In rapid
motions they do all that band or cord can perform, with the addition of
mathematical exactness, and an important saving of power. In spinning
factories these properties must be peculiarly interesting; and in
calico-printing, where the various delicate operations require great
precision of motion. In clock-making also, this property is of great
importance in regulating the action of the weight, and thus giving full
scope to the equalizing principle whatever it be. I may add, it almost
annuls the cause of anomaly in these machines, since a given clock will
go with less than 1/4 of the weight usually employed to move it. Another
useful application may be mentioned; in flatting mills, where one roller
is driven by a pinion from the other, there is a constant combat between
the effort of the plate to pass equally through the rollers, and the
action of the common geering, which is more or less convulsive. Whence
the plate is _puckered_, and the resistance much increased, both which
circumstances these wheels completely obviate; and many similar cases
might be adduced.
I shall only add, that my ambition will be highly gratified if, through
the approbation of this learned society, I may hope to contribute to the
improvement and perfection of the manufactures of this county; and if
the invention be found of general utility to my much loved country.”
Subsequently to the reading of the above paper, I had occasion to
execute many wheels on this principle; and their appearance, and use,
excited on the one hand much interest, and on the other much opposition.
I had even to complain of real injury in that contest: against which I
defended myself with a warmth that I thought _proportionate_ to the
attack.--But all this was local and temporary: and writing now for a
more enlarged sphere, and perhaps for a more extended period, I feel
inclined to lay aside every consideration, but those immediately
connected with the influence of this work on the public prosperity. I
shall therefore avoid all reference to the names either of my friends
or my opponents. My friends will live in a grateful heart, as long as
memory itself shall last; my enemies, if I have any, will be
forgiven--or, at worst, forgotten; and my System is henceforward left to
wind its way into public notice and usefulness, by its own intrinsic
merits.
* * * * *
CERTAIN OBSERVATIONS which I was induced to make on occasion of a
re-print of the above Memoir, may assist in introducing what remains to
be said on the subject. They commence thus:
The foregoing little work, which first brought this subject into public
notice in this town, was not the only method employed to develope its
principles, and urge its adoption. A second paper was read, at the next
meeting of the society, and some time after, a third, at the Exchange
Dining Room; on both which occasions new modes of reasoning were
pursued, and new kinds of proof adduced. On the first, a model was
exhibited of two screw-formed teeth (connected with proper centres)
exactly like those represented in fig. 6; by the action of which on each
other, it became manifest that teeth of this angular shape _do_ work
together without inconvenience, and therefore, that all sensible
friction is, in this case, done away.