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OSTRAND’S SCIENCE STLEIES.
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THT1
)F M<ME
TWO L.ZCTUP7:-'
RELATING TO EEULEADi ItETbO^
Mvered at Sootti Kensingtor Mcse:iti,
BY
5r>oi AI„EX. B. W. KENNEDY.C.E.
VSTX3 'T INTRODUCTIO* BY
Prof. R. H. fHURSTON, A.M.C E,
-______
REPRUITED FROM VAN HCSTRAND’S MAGAZINE.
1881.<Xr U
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flfotnell ttttitterattg tibrarg
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dtfeta boola gM^DUMd irtiiiVAN NOSTRAM SCIENCE SERIES.
No. 17.—WATER ANO WATER SUPPLY. By
Prof. W. H. Corfield, M. A., of tbe
University College, London.
No.. 18.—SEWERAGE AND SE W AGE UTILI- >
ZATION. By Prof. W. H. Corfield,
M. A., of the University College, Lon-
don.
No. 19.-STRENGTH OF BEAMS UNDER
TRANSVERSE LOADS. By Prof.
W. Allen, Author of “ Theory of
Arches.” With Illustrations.
No. 20.—BRIDGE AND TUNNEL CENTRES.
ByJoim B. McMasters, C. E. With
Illustrations.
No. 21.—SAFETY VALVES. By Richard H.
Buel, C. E. With Illustrations.
No. 22 —HIGH MASONRY DAMS. By John B.
McMasters, C. E. With Illustrations.
No. 23.—THE FATIGUE OF METALS UNDER
REPEATED STRAINS, with various .
Tables of Results of Experiments. From
the German of Prof. Ludwig Spangen-
berg. With a Preface by S. H. Shreve,
A. M. With Illustrations.
No. 24.—A PRACTICAL TREATISE ON THE
TEETH OF VVHEELS, with tlieTheo-
r.y of the Use of Robinson*s Odonto»
graph. By S. W. Robinson, Prof. of
Mechanical Engineering. Illinois In-
dustrial University.
No. 25,—THEORY AND CALCULATIONS OF
CONTINUOUS BRIDGES. By Mans-
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tions.
No. 26.—PR \ CTIOAL TREATISE ON THE
PROPERTIES OF CONTINUOUS
BRIDGES. By Charles Bender, C. E.
No. 27,—ON BOILER INCRUSTATION AND
CORROSION. By F. J. Rowan.TUE
TWO LECTURES
RELATING TO REULEAUX METHODS,
Deliyered at Soatb Kea$iH2toa Ksseiffl,
BY
Prof. ALEX. B. W. KENNEDY,C.E.
WITH AN INTKODUCTION BY
Prof. R. H. THURSTON, A. M.C.E.
REPRINTED FROM VAN N08TRAND’S HAGAZINE.
NEW YORK:
D. VAN NOSTRAND, PUBLISHER,
23 Mdb«a» ani. 27 Wabrkn Street.
1881.A. /63/
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UIMIVERSITY
^.LIBRARVV
COPYRIGHT—D. VAN NOSTRAND—1881.INTRODUCTION.
The following reprint of Professor
Kennedy’s lectures illustrating Eeu-
leaux’s new methods of treatment of the
study of the kinematics of machinery,
and the Science of pure mechanism, as
it has been termed by Professor Willis,
is published at the request of the writer,
for the purpose of placing in the
hands of every mechanical engineer
an exhibit of the methods devised by
one of the most distinguished engineers
and men of Science in Europe for the
study of the motions of connected parts
of machinery. It is hoped that these
short and well-considered lectures may
assist the student in the effort to compre-
hend those new methods, and to make
use of them in professional work, and
especially that they may have a wider
range of influence, and may lead to a1Y
more general introdaction of Reuleaux’s
treatise*—Theoretische Kinematik—the
publication of which will probably come
to be admitted to be the most important
event in the history of kinematics. In fact,
the Science of related motions, as the
writer has called it, has never before as-
sumed that definite, logical and thor-
oughly complete form which should dis-
tinguish every tme Science.
The defects of the usual method of
studying a simple machine as a whole, and
without reference to general elementary
principies; the acknowledged want of a
defined system of study and of a fixed
nomenclature, and a natural and logical
method of expression of principies and
facts, and the difficulty met with in
securing a general comprehension, among
teachers, of the radical distinction be-
tween the two Sciences relating to mo-
tion and to forces, have been serious
obstacles to the advance of the former.
♦The Kinematics of Machinery. Outlines of a
Theory of Machines. By F. Reuleaux. Translated
and edited by A. B. W. Kennedy, C. E. London: 1876Reuleaux has done much to ciear
away these obstacles, and has made a
splendid effort to place the Science of the
kinematics of machinery in its proper
relations to the older branches of dynam-
ics. He first outlines the theory of ma-
chines, studying their various forms and
determining what conditions are common
to ali, and therefore essential. The
machine is studied as a useful and practi-
cally yaluable apparatus in the light of a
Science thus founded. The distinction
between the theoretical and the practical
departments of the study is recognized—
or rather as the able author says, between
theory and empiricism—and the en-
deavor is made to aid in the substitution
of practically applicable exact methods
for the empirical and “ rule of thumb *’
determinatione, which, although un-
avoidably adopted in the absence of
precise scientific knowledge, are glad-
ly abandoned by every intelhgent
engineer when precise and logically
correct methods become practically
available.vi
The methods of treatment of the phil-
osophy of mechanism have hitherto been
the algebraic or geometric solution of
detached problems, and the study of un-
related sets of elements. The origin of
the Science and the basis upon which it
stands have been hardly perceived, and
no attempt had previously been made
to reveal foundation, framework and cov-
ering, and to exhibit their relation to the
architectural whole, when Reuleaux be-
gan his work of analyzing mechanisms,
detecting elements, studying their com-
binations and discovering those laws of
their operation which constitute the new
Science. The resuit has been the pro-
duction of a work which contains both a
general theory of machines and a scien-
tific treatment of their movements; if
not absolutely complete in itself, this is,
at least, more complete than the earlier
treatises. The work will, in some re-
spects, pro ve disappointing to the engi-
neer familiar with every-day practice, but
the fact gives him the opportuni ty to do
a great work, by building into the frameYU
erected by the great German engineer
the detached members to be found in
Willis and in Rankine and otber less
well-known works, and thus making the
structure more complete and more valua-
ble. Reuleaux has led the way, and has
pointed out a path in which ali are at
liberty to walk as far as their ability and
their strength may permit.
Those whose time and inclination may
lead them to the examination of the
Work, to which attention is here called,
will find that the author is as systematic
in his methods, and as logical in his
statements and conclusions as Euclid,
himself, could have desired. He first
delines a machine:	“ A machine is a
combination of resisting pieces arranged
in such manner that the mechanical
forces of nature may be caused to do
Work with certain determinate motions.”
Practical mechanics has been subdi-
vided into:
The Study of Machinery in General.
The Theory of Machines.
Machine Design.vm
The Science of Pure Mechanisms.
Keuleaux would construet a single
homogeneous philosophy of the Science
of machinery ; the manner in which this
is done will be comprehended after the
reader has become acquainted with the
illustrations of his method, given by
Professor Kennedy. The basis of all
is found in the following definition :
“A mechanism is a closed kinematic
chain, which is compound or simple, and
consists of pairs of elements; these
carry the envelopes for the motion which
the parts in contact must be given, and
by these envelopes all other motions are
prevented.” When an effort is applied
to this combination, work is done, and
done in a fixed and definite way, the re
sistance being overcome through a cer-
tain predetermined path; this mechan-
ism so operated, constitutes a machine;
and, the combination existing, it remains
a machine whether actually put in motion
or not; many machines do not necessa-
rily move, although, assuming motion to
take place, the relations of effort to re-IX
sistance and of motions of parts are as
definite as if the machine were in con-
tinuous operation doing work.
The treatise of Beuleaux begins with
the study of the various motions which
oceur in mechanism, and the conclusion
is promptly reached that all relative mo-
tions of parts in any given plane—con-
plane movements as he calls them—may
be considered as rolling motions, and that
they become determinable by the study
of their “ centroids,” i. e., of the pairs
of curves which, rolling together, would
produce the motion. This treatment is
evidently very closely related to that
made so familiar by Bankine and earlier
writers, in whose methods instantaneous
axes are taken as the centers of such
revolution. As stated by Beuleaux:
“ Every relative motion of two con-plane
bodies may be considered a cylindrical
rolling, and the motions of any points in
them may be determined so soon as their
cylinders of instantaneous axes are
known.” Each line of coincidence, or
of contact of the elements, of these roll-X
ing curved surfaces is an “ instantaneons
axis of rotation.” Only one pair of
these centroids can exist for any one rel-
ative mofcion.
Seeking these curves for cases of mo-
tion not confined to one plane, it is found
that motion about a point may always be
considered as a “ conic rolling,” the
cones being generated by the instantane-
ous axes determined by the given motion.
Proceeding stili further, a more general
—the most general—case is studied and
it is found that “ all relative motions of
two bodies may be considered as twist-
ing or rolling of ruled surfaces, or “ ax-
oids.” Pairs of ruled Surfaces may be
made to exhibit allpossible motions.
Where the ratio of twist to transla-
tion becomes indefinitely small, the mo-
tion becomes that of sliding; when in-
finitely great, it becomes simple revo-
lution. The detection and study of
these axoidal surfaces is as attractive
and interesting as it is useful, and the
student becomes quickly familiar with
this method of representation of a mech-XI
anism, and soon learns to see the ma-
chine as a set of pieces playing their rel-
ative parts in the whole, in direct rela-
tion with the invisible geometrical, and
always graceful, forms of the “ axoids ”
which—and not the frame and solid
portions of the structure—really consti-
tute with them the real machine. The
laying out of the common teeth of
wheels is an example of the determina-
tion of plane centroids; hyperboloidal’
■wheels have axoids.
Elements of mechanisms become now
capable of ready classifieation,* and their
combinations are easily made known and
classified, and it becomes almost self-evi-
dent that, in the machine, the office
of the several parts is to secure the rela-
tive motions appropriate to each pair of
elements and perfect restraint against
all other movements. The laws of such
restraint are simple, and are easily dis-
covered and readily enunciated. Thus,
* ^luids and flexible and elastic bodies are, as a mat-
ter of course, elements of any machine in which tbey
fcppear.xii
the study of mechanism becomes truly
scientific, and the examination of de-
tached and special problems, illustrated
by the methods of Willis, and more
markedly by Rankine, becomes seconda-
ry to this primary general treatment.
As remarked in the beginning, a sys-
tematic notation and the development
of a system of scientific expression of*
the laws of machine movement, in which
feuch a notation is used, constitutes
a grand feature of this work. Earlier
attempts, by Babbage and others, have
been barren of valuable results, simply
because the scientific foundation was not
first laid down; and, in less degree,
because of their imperfect nature. It is
wonderful that this later system, once
developed, should be found so simple.
A dozen primary symbols designate all
the parts used in machinery; four sym-
bols indicate forms, and two identify
liquid and gaseous elements, while some
thing more than a dozen of the familiar
algebraic symbols find appropriate ap-
plication in exhibiting relation of parts.Xlll
These symbols are simply and relatively
combined to represent any machine, and
a number of convenient contractione
render their use more convenient in the
more frequently occurring cases. The
special formulas thus constructed as rep-
resentative of particular machines, and
the general formulas representing classes,
are constantly used in analysis, and in
synthesis as well, and the identity of
visibly differing machines, or the radical
differences between mechanisms seeming
to the eye almost- identical, become at
once evident when the material disguise,
in which they appear to the untaught
mind, is stripped off, and they become
known only through their kinematic, or
true mechanical, relations.
This last point is best exemplified by
the analysis of “slide crank-trains,” in-
cluding “ quick-return motions ” of crank
trains generally, and especially and
most beautifully in the study of “ rotary
engines,” in which familiar motions
take a peculiar disguise while retaining
all their original kinematic characteris-XIV
tics. For example:	Galloway’s rotary
steam engine is shown to be simply an
awkward combination of steam “ cylin-
der ” with Watt’s planet-wheel rotation.
An analysis of complete machines, and
an historical retrospect brings ont very
strongly the fact wliich every engineer
must have remarked: that attempts to
follow the kinematics of Nature have
almost invariably been misleading to the
machine designer, and that the most
rapid progress has usuallv followed when
the inventor has broken away from this
seductive but fettering idea, and has
boldly and independently sought to at-
tain a definitely known object, by the
most direct and simplest of familiar
mechanical devices.
Reuleaux’s work concludes with a
treatise upon the most important and
highest—and naturally least developed
—part of the newly constructed Science:
that of scientific kinematic synthesis, or
the building up of a machine designed
to do a specified work, and to have cer-
tain required motions. Reuleaux wouldXY
substitute this for that blindly groping
for, or guessing at, results which so gen-
erally leads to invention. He would
make the process of invention a scientific
one, and would thus raise the inventor
to a higher plane, and give him the in-
creased distinction which is due a higher
order of intellectual work.
This may be illustrated thus:	In the
design of a cotton-combing machine,
which once feli into the hands of the
writer, certain objects were to be accom-
plished, and certain means were availa-
ble. The uncarded cotton was to be re-
ceived upon an apron, carried forward to
a point wbere it could be seized by the
mechanical fingers, which were to pre-
sent it to the combs, holding it while one
end of the mass of fibers was operated
upon; this operation completed, another
set of fingers were to seize the combed
ends of the fibers, now released by the
first set, and were to present the un-
combed ends to the combs. Both ends
of the fiber being finally combed, the
cotton was to be deposited carefully andXVI
undisturbed upon the apron, to form the
lap, which was then removed from the
machine.
In designing this machine, each opera-
tion was studied with a view to deter-
mine what “ kinematic chain ” was best
adapted to do that work, with the exact
motion demanded, and this chain being
obtained, it was located properly, and
the next similarly worked out. When
these combinations of motions were final-
ly grouped in snch manner that each
succeeded its leader in correct order and
in proper relation, the form of each
piece or “element” was worked out;
when ali the elements had been given
shape, and found to admit of the re-
quired cycle of kinematic changes with-
out interference, the “ centroids ” of
their motions were studied to obtain the
best means of connecting ali to a com-
mon fixed “link ”—the frame of the ma-
chine. The aprons were sliding pieces ;
the feeding rollers were rotating ele-
ments ; the mechanical fingers were vibrat-
ing pieces carried by parts themselves vi-XVII
brating, and the combs were simple “ slid-
ing elemenfcs.” The centroids determined
by the motions of the nippers or fingers,
gave the form of the cams needed to
produce those motions; the cams actuat-
ing the other intermittently or irregu-
larly moving pieces were laid out in the
same manner. The machine now con-
sisted of one extended kinematic chain,
operated from one end. The fixed points
in the chain, and those centroids which
were themselves fixed in space, now gave
location to parts of the frame of the
machine adapted to their support, and
the shape and proportion of the frame
became readily determinable as soon as
the lines of effort at these resistant
points were laid down. The last part of
this problem in machine design was
solved when a frame had been fitted to
the set of fixed points, having proper
form and size to sustain all stresses Corn-
ing upon them, and so arranged as to
permit every motion to take place with-
out bringing moving parts into collision
with its own members. The frame wasXVI11
laid out in skeleton form, and when
found suitable in all respects, was drawn
in full and all details worked in.
Such a process of “ kinematic synthe-
sis ” Reuleaux would adopt in all cases,
and the construction of kinematic for-
mulas would be the first steps to be
taken in his method of “ scientific inven-
tion.” The other steps would follow as
just indicated; the selection of the
needed elements would be facilitated by
that reduction to simplest terms of every
proposed machine, which would become
so easy when thus presented to the
mind, uncomplicated by conceptions of
conventional form.
The synthesis of a kinematic Science,
which we have been able to follow, must,
in the opinion of the writer, ultimately
prove of real Service to the profession.
Nearly all great advances of this charac-
ter, like nearly all great mechanical in-
ventions, are a generation, at least, in
taking their place in the world; it is to
behoped that this may prove an exception
to the rule. It will become evident atXIX
the first glance that the method of Reu-'
leaux does not render valueless the work
of Willis, of Rankine or of Redten-
bacher.
The former should be studied as a
basis for the detailed work of the others;
it is a general Science of machinery,
which, once rendered familiar to the stu-
dent, makes his later work easier and
more fruitful of practical resuit. He
should become familiar with this scien-
tific basis, and should then study the
relation of parts and of motions in de-
tail with Willis and Rankine, and should
finally make himself familiar with the
complete machine, which he is likely to
find useful in his professional work, and
endeavor to become capable of design-
ing them, and of synthetically produc-
ing new forms of mechanism.THE K1NEHATICS OF MACHINERY.
Most of tbe models used to illustrate
this and the following lecture belong to
the Kinematic Collection of the Gew-
erbe-Akademie in Berlin, and have been
designed by Professor Reuleaux, who is
the Director of the Academy and a Pro-
fessor in it. The rest were sent to the
Loan Collection by Messrs. Hoff and
Voigt of Berlin, and Messrs. Bock and
Handrick of Dresden. In essentials
there is no difference between the Berlin
and the Dresden models. Both have
been designed specially for use in instruc-
tion in the Kinematics of machinery.
I must first try to explain briefly,
but exactly, what I mean by the phrase
“Kinematics of machinery.” Professor
Reuleaux, whose models are before us,
delines a machine as “a combination of
resistant bodies so arranged that by their
means the mechanical forces of nature22
can be compelled to do work aecompa-
nied by certain determinate motions.”
The complete course of machine instruc-
tion followed in some of the Continental
teehnieal schools covers something like
the following ground:
First, there is the perfeetly general
study of machinery, technologically and
teleologically. Then there comes what
we may call the study of prime movers,
which in terms of our definition would
be the study of the arrangements by
means of which the natural forces can
be best compelled to do the required work.
Then comes the study of what may be
called “direct actors,” or the direct-act-
ing parts of machinery; in the terms of
our definition, the arrangement of the
parts of a machine in such a way as best
to obtain the required resuit. Next comes
what we call machine design; the giving
to the bodies forming the machine the
requisite quality of resistance. Machine
design is based principally on a study of
the strength of materials.
One clause of the definition stili re-23
mains untouched. The machine, we
said, does work accompanied by certain
determinate motions. Corresponding to
this we have in machine instruction the
study of those arrangements in the ma-
chine by which the mutual motions of its
parts, considered as changes of position
only, are determined. The limitation
here must be remembered; motion is
considered only as a change of position,
not taking into account either force
or velocity. This is what Professor
Willis long ago called the “science of
pure mechanism,” what Rankine has
called the “ geometry of machinery,”
what Reuleaux calls “kinematics,” and
what I mean now by the “ kinematics of
machinery.”
Ths results of many years’ work of
Reuleaux in connection with this subject
are embodied in his book Die Theoret-
ische Kinematik, which I recently had
the pleasure of translating, and I shall
endeavor to give you an outline of his
treatment of the subject. It cannot be
more than an outline, as you will readily24
understand. The subject is a very large
one, and I have had to choose between
taking up many branches of it and merely
mentioning each, and confining myself
to a few points, and going more into de-
tail about them. I have chosen the lat-
ter plan, believing that the former would
be of little benefit to anybody. It will
be easy for those who are sufficiently in-
terested in the matter to follow it up,
and to study those parts whieh I omit,
by the aid of the book I have just men-
tioned. My lecture to-day will be prin-
cipally theoretical, and to-morrow I shall
go more into practical apphcations. So
far as possible, as I have Professor Reu-
leauxs models before me, I shall endeav-
or to follow his own order in treating the
subjeci
I presume you are acquainted, to a
certain extent, with the ordinary method
of studying “ pure mechanism; ” the
method originated by Monge (1806), de-
veloped in Willis’ well-known Principies
of Mechanism (1841), and made popu-
lar, to a great extent, by Prof. Goodeve’s25
capital little text book and others. Each
mechanism is studied for and by itself,
in general, by the aid of simple alge-
braic or trigonometric methods, and is
spoken of in reference to a certain “ con-
version” of motion which occurs in it.
Thus, we have the conversion of circular
into reciprocating motion, the conver-
sion of reciprocating into circular, &c.,
and simple formulae express certain rela-
tions between the motions of two or
more moving points. In this way we
know something important about a great
number of mechanisms, and arrive at
many results which are both useful and
interesting. Some things are stili left
wanting, however; and these things may
be summed up in this way:
(1.) We notice at once that we have
taken the mechanism as a whole. We
do not analyze it in any way whatever,
and therefore,
(2) We have scarcely any knowledge
of its relations with other mechanisms,
or (what is quite as important) of the
various forms which one and the same26
mechanism may take. We shall see pre-
sently how extraordinarily various these
forms are, We have never a general
case with special cases derived from it;
each case is treated by itself as a special
one. Then
(3) The mechanism is studied in gen-
eral from a point of view which gives ns
only the eonditions of the motion of two
points in it, or two portions of it, and is
then left. The kinematic eonditions of
the mechanism as a whole remain abso-
lutely untouched.
In such a mechanism as that of an or-
dinary steam engine, for instance, we
study the relative motions of the guide
block and the crank, or, I ought, per-
haps, to say of the axes of the cross
head and of the crank pin. We thus
know the motions of two points in the
rod which connects those axes, the “eon-
necting rod,” but we leave the motions
of its other points untouched. It may, of
course, be said that these others are of
much less practical importance. This is
true to some extent, although their practi-27
cal importance is greater than might be
supposed at first. But in any case these
motions must certainly be studied if we
are to obtain a complete knowledge of
the mechanism to which tbey belong.
Any method of study, therefore, which
covers all the kinematie conditions of the
mechanism, instead of the mechanical
conditions of two or three points only,
possesses in that respect very great ad
vantages.
The treatment of mechanisms which I
shall sketch to you, is intended to rem-
edy some of the defects which I haye
enumerated. Those of yon who have
studied modem geometry, side by side
with the old methods, will recognize that
these defects are somewhat analogous to
those of Euclidean geometry. The at-
tempt to remedy thenl proceeds in lines
similar to those of modem geometry,
and will eventually, I believe, when more
fully worked out, take the same position
in its own subject.
Let us, then, look first at the analysis
of mechanisms. This is none the less28
important a matter that its results are
so very simple in many cases. A ciear
understanding of those elementary mat-
ters is of great assistance in clearing up
difficulties which occur in the more ad-
vaneed parts of the subject.
In a machine or a mechanism of any
kind the motion of every piece must he
absolutely determinate at every instant.
It will be remembered that we are at
present considering motion as change of
position only, not in reference to velocity.
The motion of change of position may
be determined by the direction and mag-
nitude of all the extemal forces which
act on the body; the motion is then said
to be free, but it is obviously impossible
to arrange such a condition of things in
a machine. The motions may, however,
be made absolutely determinate inde-
pendently of the direction and magni-
tude of external forces; and in order that
this may be the case, the moving bodies,
or the moving and fixed bodies as the
case may be, must be connected by suit-
ahle geometric forms. Motion, under29
these cireumstances, is called constrained
motion.*
If I allow a prismatic block to slide
down the surface of an inclined plane its
motion will be free; it is determined by
the combination of external forces which
act upon the block. If the block be
pressed on one side as it slides, it at
once moves sideways, and can only be
kept in a straight path if directly the
pressure is exerted on the one side an
equal and opposite force (or a force
which has a resultant with the first in
the direction of motion) be caused to
act upon it on the other. If, on the
other hand, the block be made to slide
between accurately-fitting grooves (like
a guide block in a machine), inclined at
the same angle as the plane, and like it
fixed, the block may be pressed sideways
or in any other direction, but no altera-
tion in its motion can take place; the
motion is “constrained,” it can occur
♦Essentiallyitdoes not differ from free motion ; the
differenee really lies in the snbstitntion of stresses or
molecular forces, which are under our complete con-
trol, for external forces.30
only in the one direction permitted by
the guiding grooves. In the one case
the extemal force has to be balanced by
another extemal force; in the other the
balancing force is molecnlar, i. e., is a
stress and not an extemal force, and
comes at once into play the instant the
disturbing force is exerted. The geo-
metric forms whieh are used in this way
to constrain or render determinate the
paotions in machines are very various, and
are chosen in reference to the particular
motion required. If every point in a
body be required to move in a circle
about some fixed axis, a portion of the
body is made in the form of a solid of
revolution about that axis, and this is
caused to “ work in ” another similar
solid; the two forming the familiar pin
and eye. If ali points of a body be re-
quired to move in parallel straight lines
we get, similarly for guiding forms, a pair
of prisms of arbitrary cross section; a
slot and block. If every point of a
body be required to move in a helix of
the same pitch we use a pair of screws31
of that pitch, one solid and one open, for
constrainingthe motion—a screw andnut.
The general condition common to
these yery simple forms is that, in each
case, the path of every point in the mov-
ing body is absolulely determined at
every instant, that is to say, the change
of position of the moving body is abso-
lutely determinate.
The geometric name for these mutu-
ally constraining bodies is envelopes, and
each one is said to envelope the other.
We shall call them (kinematic) elements,
and the combination of two of them we
shall call a pair of elements.
Those we ha ve mentioned are special
and very familiar and important cases of
pairs of elements, which are of great
simplicity. They have the common
property of surface contact, the one en-
closing the other, and are therefore
called closed or lower pairs of elements.
They are, moreover, the only closed pairs
which exist. They are, further, the only
pairs in which all points of the moving
element have similar pairs.32
Every point of an eyer for instance,
moves in a circle about the same axis.
If there were attached to it a body of
any size or form whatever, all its points
would move about the same axis. The
“ point paths ” would all be coneentric
circles. Again, whatever the extemal
size or shape of a nut, every point in it
moves in a helix of the same pitch about
the axis of the screw; the point paths,
that is, would be similar.
The genera! condition of determinate-
ness of motion can, however, be fulfilled
by an immense number of other pairs of
elements. The theory of these is too
large a subject to be entered into just
now, I must merely direct your attention
to the existence of such combinations.
Fig. 1 represents one of the simplest
that can be used. Here one of the ele-
ments is an equilateral triangle, ABC,
the other is the u duangle ” RPSQ*
The latter moves within the former,
touching it always in three points, or
rather along three lines. Its motion is
just as absolutely determinate as the33
motion of a pin in an eye. It is free to
move at any instant only about the point
in which the three normals to the tri-
angle at the points of contact intersect
(as Q in the Fig.). The models before
you show a few of the many forms taken
by such pairs of elements. It is worth
A
while noticing a few points in which the
motions determined by them differ
from the motions of the closed pairs.
First, as we have already seen, the con-
tact of the elements determining the
motion was surface contact in the former34
case, while here it takes place only along
a finite number of lines. Then the mo-
tions of all points in the first case were
similar; in these pairs the motions of
the points are not similar, but entirely
dissimilar, the motion of each point de-
pending entirely upon its position. Fig.
2 shows a few of the point paths of the
pair of elements shown in Fig. 1. The
strikingly different curves obtained from
one pair of elements, according to the
choice of the describing point, is too
obvious to need further notice.*
These pairs of elements are called
higher pairs.	They have only a few
applications in practice, their interest
being chiefly theoretical. From our
present point of view their theoretic in-
terest is considerable, because of their
exact analogy with the lower pairs.
There is another difference between
the two kinds of pairs which deserves
notice, for reasons which will be better
* The triangle UTQ and the three curve» wlthin it,
which have Mi for their center, are point paths. The
curve triangle and the duangle shown in thicker lines
will be explained further on.I
Fig.235
understood afterwards. The pair of
elements determine the relative motion
of the two bodies connected by them. If
one body be stationary on the floor or
the earth, the moving body has the same
motion relatively to the floor or earth
that it has to the other element. If I
move about both bodies in my hand,
both have motion relatively to the earth,
but the relative motion of the one to the
other remains unchanged. It is of
course only a case differing in degree
from the former one, for in the former
one both bodies had the motion of
the earth itself, while one had the addi-
tional motion which I gave it. We may,
however, not to be pedantic, speak of
anything as “ fixed,” or “ stationary ”
which has the same motion as the earth.
Now, (in this sense) we may fix either
element of a pair, and with the lower
pairs the relative motion taking place
remains the same whichever element
be fixed. With the higher pairs, on
the other hand, the relative motion is
altered, and the point paths become en-86
tirely different. The point paths of the
duangle relatively to the triangle are, for
instanee, quite different from those of
the triangle relatively to the duangle.
This change of the fixed element is
called the in ver sion of a pair.
The ultimate resuit of our analysis of
mechanisms is then pairs of elements;
we cannot go below this. The pairs we
have noticed are of two kinds. each hav-
ing their own definite characteristics.
If, now, two or more elements of as
many different pairs be joined together
we get a combination whieh is ealled a
(kinematic) UnJc. It is obvious that the
form of sueh a link is, kinematically, ab-
solutely indifferent. The choice of its
form and material belongs to machine
design. It may be brick and mortar,
cast iron, timber, as we shall see after-
wards, but the fact that this is indiffer-
ent, kinematically, cannot be too dis-
tinctly kept in mind.
We can make combinations of links
by pairing the elements whieh each con-
tain to partner elements in other links, and37
such combinations are called kinematic
chains. Thus, if we denote similar ele-
ments by similar letters, aa, bb> ec, &c.,
and the link connection by a line, we
may indicate some of the chains obtain-
able from four pairs and four links, thus:
a----bb------cc---cici----a
(we suppose the “ chain ” to return on
itself and the two elements a to be
paired, the whole forming a closed
chain); or,
a----cc------bb-----dd----a
or
a----dd------cc-----bb----a &c.
For the sake of illustration we give in
Fig.3
Fig. 3 a sketch of a familiar chain con-
taining four links, each connected to the
adjacent link by a cylinder pair of ele-38
ments. The axes of the four pairs of
elements are parallel.
We have, then, in the kinematic chain,
a eombination so constructed that ali its
parts have determinate motions, motions
absolutely fixed by the form of the ele-
ments earried by its links, and independ-
ent (considered as changes of position)
of the application of external foree. To
converfc the chain into a mechanism we
have only to do what we have already
done in connection with pairs of ele-
ments, fix one element—or, as each ele-
ment is rigidly connected with a link,
we may say preferably fix one link. Any
link may be fixed, the chain, therefore,
gives us as many mechanisms as it has
links. In general these are different, in
special cases only two or more of them
are the same. We shall be able to enter
into this part of our subject at some
length in the next lecture; at present it
wiU suffice to note two or three of the
leading characteristics of chains and
mechanisms which we can now easily
recognize. These are,39
(i.) That the motion of any link rela-
tive to either adjacent link is determined
by the pair of elements connecting
them.
(ii.) That the motion of any link rela-
tive to any other than its adjacent links
depends on ali the elements of the
chain.
(iii.) That no link of a mechanism can
be moved without moving all the other
hnks exeept the fixed one, and
(iv.) That there can be only one fixed
link in a mechanism.
The two last propositions require a few
words of explanation. Suppose that in
any combination of, say, four links, two
can be moved without moving the other
two, the combination is actually one of
three links only, for clearly the two im-
movable links may be made into one, and
are two only in name. This is very often
the case in machinery, where special me-
chanisme are frequently used for the
express purpose of connecting rigidly
two or more links, and making them act
as one, at certain intervals.40
If, however, in tlie combination sup-
posed, one link be fixed, while two can
be moved and tliefourth can either move
or be stationary, the combination no
longer comes under our definition of
constrainment, for the motions are at a
certain point indeterminate, at the point,
namely, when it is possible for the fourth
link either to move or to stand. Chains
often occur in which this would be the
case, were it not that mechanicians take
means, either by adding other chains or
in other ways, to constrain the motion
which would otherwise be useless to
them.
We ha ve now obtained some idea of
the way in which mechanisms are formed,
of the elements of which they consist.
Before applying the knowledge we have
thus acquired I must direct your atten-
tion to some geometric propositions
which will greatly facilitate the theoretic
dealing with these mechanisms.
In order that I may not enter into too
wide a subject, I shall confine myself
here to the eonsideration only of “ con-41
plane ” motions, or motions in which ali
points of tlie moving body move in the
same plane or in parallel planes. The
limitation is a large one, but the cases
included under conplane motion cover
the greater part of those which occur in
practice. The method which I have to
describe is equally applicable to general
motion in space as to simple constrained
conplane motions of which I shall speak.
Let me remind you that the motion of
any figure moving in a plane is known if
the motion of any two points (i. e. of a
line) in it be known. The motion of any
body having conplane motion is known
if the motion of a plane section of it,
parallel to the plane of motion, be known.
Such a plane section ,of it is, of course,
simply a plane figure moving in its own
plane. The motion of any body having
conplane motion (as in nine cases out of
ten in machinery) can, therefore, be deter-
mined by the determination of the mo-
tion of two points. In speaking now,
therefore, of the motion of a Une for
shortness’ sake, it must be remembered1
42
that we are really covering all eases of
conplane motion of solicl bodies.
In Fig. 4 PQ and P^ are two posi-
tions of the same plane figure, or plane
seetion of a body having conplane mo-
tion. If now we have two positions (in
the same plane) of any plane figure, we
know that the figure can always be
moved from the one to the other by tum-
ing about some point in the plane. The
position of the point O, about which the
figure can be turned from the position
PQ to the position P^Qj can be found at
once by the intersection of the normal
bisectors to PP, and QQt. The motion
of PQ in the plane is, of course, its mo-
tion relatively to the plane, and there-
fore relatively to any figure (as A B) in
the plane. Such a point O as we have
found here is called a temporary center,
because the tuming or motion takes
place about it for some finite interval of
time. It will be remembered that not
only the two points and PQof the figure,
but every other point of it, must have a
movement about* this same point O at the43
same time. Now suppose we have some
further position of the same figure, as
for example at the position marked P„Qa,
we can find in the same way the center
about which the figure must be turned to
move from P1Q1 to P,Q3. We may indi-
cate this point as Oj. Similarly taking
other positions of this figure PSQ3 and
so on, we can find other points, OaOa,
&c. By joining the points OOjOjOj, we
obtain a polygon, and if the figure in its
motion come back to its original position
the polygon also comes back on itself,
and passes again through the point O.
Such a polygon, whether it be closed m
this way or not, is called a Central poly-
gon; its comers are the temporary cen-
ters of the motion of the figure.
I have pointed out that ali the points
in the figure PQ move round O during
the motion from PQ to PxQt. They
move round O necessarily through some
particular angle, the angle POP,, and
every point moves through the same
angle, which we may call (px. As the
figure may have any form we choose, let44
us suppose it so extended as to contain
a line which is the same length as 001?
and which makes with 001 the angle cpx,
that is to say, the angle through which
the figure moves about O. Such a line
is shown in Fig. 4 by MMr
We have, then, a line	forming a
part of the figure PQ, equal in length to
001? the points O and M coinciding, and
the angle 0,MM1 being = q>v Then
when the figure has completed its mo-
tion about O, MM, and 00, must coin-
cide. Take further similarly MjM^O,
02 and so placed that when coincides
with O,, < 02MjM2 = <p^ then when the
figure takes its third position, complet-
ing the turning about Oj? M,M2 coincides
with 0,09. Similarly we can obtain Ms
M4, &c. The figure thus found is another
polygon, which we may call a second Cen-
tral polygon.
These polygons have important prop-
erties, the principal of which can be very
easily recognized. The first polygon
does not alter its position during the
motion of the body; it is therefore fixed,45
so that it may be considered as a part of
any figure such as AB, which is fixed or
stationary in the plane of motion. The
second polygon moves with PQ and
forms (by construetion) part of the same
figure with PQ. This second polygon
then, by the consecutive turnings of its
comers upon the corresponding corners
of the first (and equal-sided) polygon,
will give to PQ the required changes of
position relatively to the fixed plane or
to the figure AB lying in it.
If, therefore, we know the Central
polygons for the given motion, we know
not only the changes of position of the
points P and Q, but those of every other
point connected with the moving figure,
whatever form it may have. For at any
one instant every point in the figure is
moving about the same center. In
studying the relative motions of the
figures we may, therefore, quite leave
out of sight their form if we only know
the central polygons for the motion.
These teli us, so far, all about the mo-
tion which is taking place.46
We may go further, however. We
have recognized the fact that the relative
motion of two figures or bodies may take
place equally whether one or the other
of them be fixed, or both moving. In
the case before us we have supposed AB
fixed and PQ moving relatively to it.
The second polygon then moves on the
first, and expresses the relative motion
taking place. If, however, we suppose
PQ fixed and AB moving, then the poly-
gons stili express the relative motion;
but the second is now fixed and the first
rolls upon it. This follows directly from
the constitution of the polygons. The
properties of the polygons as expressing
the relative motions of the bodies to
which they belong are, therefore, re-
ciprocal.
You will have noticed, no doubt, that
the polygons do not express continuous
motion. They define only a series of
changes of position in their beginning
and end, not telling us of the intermedi-
ate stages.
We may, however, take the consecu-47
tive position of the figures as close to-
gether as we like. The closer together
they are taken the shorter become the
sides of the polygons. If at last the
distances PPt, P^, QQt, &c., be taken,
infinitely small, each corner of the poly-
gon will be infinitely close to the next
one. That is to say the two polygons
will become curves, and of these curves
“infinitely small parts of equal length
continually fall together after infinitely
small rotations about their end points.”
In other words the two curves roll on
one another during the continuous al-
terations in the relative position of the
two £gures. Instead of finding points
now by the intersection of normal bi-
sectors, they are found by intersection
of normals to the paths of P and Q (Fig.
5). The tuming about each point now
occurs (not in general) for a finite period,
but for an instant only. Each point is
therefore ealled an instantaneous center.
The curve containing ali the instantane
ous centers, or the locus of instantaneous
centers, is ealled a centroid. Without48
giving them any special name, several
writers on Mechanics have made more or
less use of these curves. Among these I
may mention Dwelshauvers-Dery, Schell
and Proli. Reuleaux has, however, given
them a name (Polbahnen), and has
made some special use of them, more,49
I think, than has been made by former
writers.
While the polygons only represent a
series of isolated positions of a body, the
eentroids, rolling on each other, repre-
sent the whole motion eontinuously.
Like the Central polygons their proper-
ties are reciprocal. If then the eentroids
of two«figures be known, their relative
motions for a series of changes of posi-
tion, each infinitely small, are also known,
i. e., their motions are completely deter-
mined.
If AB and its centroid be fixed, and
the centroid of PQ rolled upon it (Fig.
5), we have now the means of determin
ing the path of motion of every point in
the Fig. PQ relative to AB, wbatever
may be the form of PQ. It is sometimes
of great convenience to be able to find
the motions of all points in a body in
such a very simple way. Reeiprocally
we can determine the point paths of A
B relatively to PQ, which, in general,
differ entirely from those of PQ rela-
tively to AB.50
If both figures be moving, as frequently
happens in practice, both centroids are
also in motion; their motion relative to
each other, however, remains unaltered.
They stili roll on one another, and their
point of contact is stili the instantaneous
center of the motion of each relatively
to the other. Each figure moves, rela-
tively to the other, about thia point,
which being common to the two cen-
troids, is common to the two figures.
They might, therefore, for the instant, be
connected at that point by a cylindric pair
of elements. There are many problems
of which the solution is greatly simpli-
fied by the recollecti on of this fact.
The point in each figure which coincides
with the instantaneous center, has, there-
fore, no motion relatively to the other
figure. We have already seen this in the
special case where the one figure is sta-
tionary, for then the point in which the
moving centroid touches the fixed one
is, by hypothesis, also stationary for the
instant; in other words, it has no motion
relatively to the fixed centroid. We now51
see the general condition of which this is
a special case.
Fig. 6 shows the centroids for the
higher pair of elements of Fig. 1. The
curve triangle UTQ is the centroid of
the triangle ABC, and the shaded duan-
gle PYQW is the centroidof the duangle
BPSQ.* As the duangle moves in the
triangle (the elements sliding upon each
other), its centroid rolls witliin the cen-
troid of the triangle. Both centroids are
* These centroids are shown on a larger scale, apart
from the elements to which they belong, in Fig. 2.52
in this case formed of ares of circles, and
ali the point patiis (being determined by
the rolling of one circular are upon
another) are combinations of trochoidal
ares.
The centroids of kinematic chains are
generally of greater complexity than
those of the pairs of elements just men-
tioned, but in some cases are quite as
simple. In Fig. 7, for example, is shown
a mechanism familiar to engineers, in
which a crank a drives a reciprocating
bar c by means of a block b working in
a slot. The centroids defining the rela-
tive motions of the links a and c are the
two circles shown in full lines, one dou-
ble the diameter of the other. These
two circles both move as the mechanism
works (supposing the link d to be fixed),
but always so that they roll eontinuously
one on the other. If instead of fixing d
the crank a were made the fixed link, the
same centroids would stili express the
relative motions of a and c. The smaller
circle, the eentroid of a, would be sta-
tionary along with the link to which it53
belongs, and the other would roll on it,
the instantaneous center for the motion
of the link c being ahvays at their points
of contact. This mechanism (a being
fixed) is used in 01dham’s coupling, in
elliptic chucks, &c. Knowing these cen-54
troids we know ali about the motions of
the two corresponding links in the me-
chanism, not only abont the motions of
some particular points in these links.
The centroids of kinematic chains can
in general be very easily determined.
Once fonnd they make us independent to
a great extent of trigonometrie or alge-
braic formulae, and enable us to deter-
mine ali we wish to know by purely geo-
metric graphic constructions. For tech-
nical purposes, at least, this is frequently
an immense advantage. There are very
few cases in which it is not more con-
venient for the engineer to employ a con-
struction than a formula, if both give
him the same resuit.
Before looking at the centroids of
other mechanisms, it is necessary to ex-
amine one particular case which often
occurs. Suppose that the lines PPj and
QQj in Fig. 4, or the tangents to the
curves at P and Q in Fig. 5, had been
parallel. It is obvious that the normal
bisectors in the one case and the normals
to the curve in the other then become55
also parallel, or, as it is for some reasons
more convenient to express it, would
meet at an infinite distance. The tem-
porary center in the one case and the
instantaneous center in the other are at
infinity. A centroid may, therefore, con-
tain one or more points at an infinite
distance, may have, that is, one or more
infinite branches. This constantly oc-
curs in mechanism, and in some cases
every point in the centroid is at an in-
finite distance. This is, liowever, a
special case; its treatment does not offer
any practical difficulty, but I cannot do
more than mention its existence here.
The centroids of the connecting rod
and frame of the ordinary steam engine
driving mechanism (the links b and cl of
Fig. 8) may serve as an illustration of56
this. When the crank a is at right an-
gles to d, the normals to the paths of
the two points 2 and 3 are parallel. The
instantaneous center of b relatively to d
is, therefore, at an infinite distance.
Eaeh centroid has, therefore, a pair of
infinite branches.
We may look, in eonclusion, at one
other case which possesses some special
interest on account of the form taken by
the centroids. It is shown in Fig. 9.
The chain contains four iinks and fonr
parallel cylinder pairs. The alternate
Iinks are equal, and the two longer Iinks
are crossed so that the chain forms an
“ anti-parallelogram ” in every position,
the angle at 2 being always equal to that
at 4, and the angle at 1 to that at 3. If
the link d be fixed, the Iinks a and c be-
come two cranks which revolve in oppo-
site directions with a varying velocity
ratio. The centroids of b and d are a
pair of hyperbolae having their foci at 2
3 and 1 4 respectively. The one rolls
upon the other as b moves, the instan-
taneous center in the position shown *be-■Ols.57
ing at the point of contact O, which is
the point of intersection of 1 2 and 3 4.
The centroids of the two shorter links
are the two ellipses which are shown in
dotted lines. They are confocal with
the hyperbolae, and their point of con-
tact is always at the intersection of 1 4
and 2 3. Their form shows at once that
the rotafcion of the axes 1 and 4 is pre-
cisely the same as that which would be
communicated by a pair of elliptic spur
wheels having the centroids for their
pitch ellipses.
In this mechanism, as in some of the
others illustrated, the centroids of two
adjacent links, as a and d or b and c, are
simply a pair of coincident points which
roll upon each other. They form thus a
limiting case of centroids, but every the-
orem which applies to the more extended
centroidal curves applies also to these
points, as can easily be seen on exami-
nation.58
LECTURE II.
We shall in the present Lecture exam-
ine in some detail a few of the results
which can be obtained by treating meclr
anisms upon the plan which Reuleaux
has proposed, and which is illustrated by
his models; that is to say, by the ana-
lytical treatment of which we have
already seen the general nature.
We have seen how kinematic chains
are built up from pairs of elements and
links. The pairing and the linkage ren-
ders the relative motions in the chain
absolutely determinate, and the deter-
minate relative motion exists equally
whether or not any link of the chain be
fixed relatively to the earth or to any
portion of space that we choose to treat
as stationary.
We have now to consider in more de-
tail the effect of fixing one link of the
chain. In practice, of course, one link is
always fixed, or in other words, its mo*59
tion relatively to the earth, to a locomo-
tive or whatever it may be, is made zero*
A chain with one link fixed is simply
what we know as a mechanism.
In examining pairs of elements we saw
that we could fix either element of the
pair with lower pairs, the relative motions
remaining unaltered; with the higher
pairs the inversion gives us a totally dif-
ferent motion. We have seen also that
we can fix any one link of a kinematic
chain just as we can fix either element of
a pair. We therefore can get as many
mechanisms from any chain as it has
links. From any such chain as Fig. 3,
for instance, which has four links, we can
get four mechanisms. The fact that a
kinematic chain gives us as many mech-
anisms as it has links appears, looked at
from this point of view, a mere matter of
course. It has, however, never been
hitherto distinctly recognized, so far as I
know, and it can hardly be realized too
distinctly, the consequences which resuit
from it being most important, as we shall
see. AII that I shall attempt to do in this60
lecture will be to look at some of tbe
mechanisms obtained from tbe particular
cbain just mentioned, and various modi-
ficatione of it.
We have already noticed tbat tbe cbain
has four links. We see furtber tbat it is
a chain in which ali tbe motions are con-
plane, eacb of its four pairs being simply
a cylinder pair, and tbe four cylinder
pairs having parallel axes. It is so pro-
portioned that by causing one link to
swing, anotber one can be made to re-
volve. In order that we may refer more
easily to tbe links, a letter is attacbed to
each in the engraving.
For convenience sake we may also use
a short Symbol for tbis cbain (tbe one
used by Beuleaux) namely, (C"4).* Tbe
C4 within brackets stands for tbe four
cylinder pairs, the Symbol for parallel
being added to indicate tbeir relative
positions. Tbis is tbe Symbol for tbe
chain, no link being fixed. To distin-
guish tbe four mechanisms formed from
it, we shall put tbe letter whicb stands
*In words “ C parallel 4.'61
for tlie fixed link in the position of an
index after the formula. Thus we can
denote tlie particular mechanism shown
in Fig. 10, in which tlie link d is fixed,
by the formula (C"4)d. We have here
then, the first of the four mechanisms
we can get from this chain. You will
recognize it easily enough as exactly
3
similar to the beam and crank of a beam
engine. The link c is half the beam, a
the crank and b the connecting rod.
The whole mechanism is an excellent il-
histration of what I said in my last lec-
ture, that the forni of the links is indif-
ferent. If you think of the mechanism
as forming part of a beam engine, for
instance, you will see in the link d the
abstract form of what is generally a most62
complex structure, a bed piate with its
bearings, an entablature and plummer
block, cast iron columns, and in some
cases even brick and masonry. All these
are represented by the fixed link d so
far as their kinematic relations are con-
cemed.
If now we fix the connecting rod b in-
stead of fixing the link d as before, we
have the meehanism (C"4)tbr It does not
essentially differ from (C"4)d. The
crank now revolves about the pin 2
which was formerly the crank pin, and
the pin 1, which formerly represented
the crank shaft, is now the crank pin,
but there is nothing changed in the na-
ture of the meehanism. By this inver-
sion therefore we have got nothing new.
Let us now fix the link a, which was
formerly the crank (Fig. 11). We have
now the meehanism (C"4)a; it contains
precise^y the same elements as before,
and the relative motions of the links are
unaltered, but as a meehanism it is en-
tirely different. It is now a combination
frequently enough used in mills and else-63
where, known by the name of a “drag-
link coupling.” The links b and d have
beeome cranks, and one drives the other
by means of the link c.
By fixing the remaining link of the
ehain, the link c, (which we have sup-
posed to be longer than a), we have the
entirely different mechanism (C''4)e, (Fig.64
12). The two arms no longer revolve
but only swing, and the link a turns
right round once in every double swing
of d and b. This mechanism is occa-
sionally used in part of its stroke in par-
allel motions with some modifications,
but is not so well known as the others.65
The four inversions of this one ehain,
therefore, give us three different mechan-
isme. Looked at separately it is hard to
see the relation in which these stand to
each other; from the point of view
which we have taken their mutual rela-
tionship has become at once evident.
Without altering the pairing at all, we
can greatly alter the chain by changing
the relative length of its links. If we
made all the links equal we should have
a square, of which all four inversions
would give similar mechanisms. If we
make b=d and c—a we get a mechanism
which is perfectly familiar in the coup-
lings of locomotives and many other
cases. All four mechanisms are again
similar, each one consisting of a pair of
cranks revolving with equal velocities
and conneeted by a link which moves
always parallel to itself.
These mechanisms are among those
which have the peculiarity to which I
alluded yesterday, that in one of their
positions their motions are not deter-
minate. This occurs at the “dead66
points ” when a and c are both standing
in the direction of the axio of d or b.
If no means be taken to prevent it, it
is then possible to move the crank
either in one direction or the other, and
the two cranks may go on revolving in
the same direction, or may revolve in
opposite directions according to cireum-
stances.
Such an indeterminateness is, of
course, inadmissible in machinery, where
we therefore adopt the well-known
method of combining two mechanisms
of the same kind, and placing them with
their cranks at right angles, so that they
do nofc cross the dead points at the same
time. The motions are thus made deter-
minate and the cranks revolve in similar
directions. We miglit, however, wish
them to revolve in opposite directions, as
in the mechanism shown in Fig. 9. It
may be worth our while to look for a
moment at the means whieh may be used
in this case to secure the determinate-
ness of the motions in the mechanism.
To distinguish between the two cases we67
may call the former “ parallel cranks ”
and represent it bythe formula (C"a||C"2),
and the latter “ anti-parallel cranks,”
(C"22I C"2). This chain, with the link d
fixed, is shown in Fig. 13.
In the case of the jmrallel cranks
all points of the centroids of b and d are
at infinity, for they are at the intersec-
tions of the parallel links a and c. We
have already seen, however, that in the
mechanism Fig. 9 the centroids are quite
different, those of b and d being hyper-
bolae. If, therefore, when the mechanism
is brought into either dead point, where
the cranks might change from the anti-
parstllel to the parallel position, we can only
make certain that the right centroids roll
upon each other, we shall get the motion68
that we want. Fig. 13 shows an arrange-
ment by which, just in that position of
possible change, a tooth made on one
link and a recess npon the other link
gear together at a point eorresponding
to the point of contact of the eentroids.
The teeth G and H are virtually formed
upon the centroid of b, and the recesses
E and F npon that of d. At the points
where these come into gear, the two cen-
troids are compelled to roll upon one
another, just as the pitch circles of two
toothed wheels are compelled to roll on **
one another, and in this way the mechan-
ism is earried over its only indeterminate
point, and the cranks remain continuous-
ly anti-parallel and revolve in opposite
directions.
This anti-parallel chain gives us two
different mechanisms. Fig. 13 shows us
(C"2Z C'\)d. In the other mechanism
(C"9Z C"9)a the two cranks revolve in
the sarne direction with very varying
velocity ratios.
Ketuming again to the chain (C"4), it
will be seen at once that we may substi-69
tute for the pair of elements at 4 a slot
and a sector concentric witli it, as in Fig.
14. The motions remain entirely unal-
tered. By adopting this contruction,
however, it becomes possible to construet
the mechanism witliout covering with it
the center of the pair 4, i. ethe point
of intersection of the links c and d. We
can therefore lengthen these links with-
out making the mechanism incon venient-
ly large. The only constructive altera-
tion is that the slot becomes flatter as
the links are lengthened. If we lengthen
them little by little until they become
infinitely long, the curved slot becomes
straight, and its center line will pass
through the point 1. The mechanism70
modified in this fashion takes the ex-
tremely familiar form already shown in
Fig. 8. It now contains three cylinder
pairs with parallel axes ; tlie fourth cylin-
der pair has become a straight slot with
a block working in it, namely, a prism
pair. The axis of the prism pair is nor-
mal to the axis of the three cylinder
pairs, and we may therefore use the Sym-
bol (C^gP-1-) for the chain in its new
form. There are here again four links,
and therefore four inversions, and we
shall find that ali four mechanisms are
now different.
We have first the mechanism shown in
Fig. 8, and familiar by its continua! use
in direct-acting engines (C"3P-L)d. Next,
following the same order as before, we
may fix the link b9 the connecting rod of
Fig. 8. The mechanism thus obtained,
(C" P*L)6, is quite different from the for-
mer, but equally familiar (Fig. 15). To
make it more recognizible, the prism pair
4 is reversed in the figure, that is the
link c is made to carry the apen prism
and d the full one. The motion71
ously unaffected by tlie change. The
mechanism can be easily seen to be that
of the oscillating engine. The link c
corresponds to the cylinder, swinging on
fixed trunnions at 3, and the link d to
the piston rod and piston of the steam
engine. We see, then, that the relation
between the mechanisms which are famil-
iar to us as the driving trains of the
direct-acting and oscillating engines, is
simply that they are different inversions
of one and the same chain.
Let us now suppose the chain fixed
upon the link which was the crank in the
last two mechanisms (Fig. 16). This
gives us a third mechanism which entire1
ly differs from either of the two former
ones. It is quite familiar as a “quick-72
retura ” motion in some machine tools,
for which purpose also the mechanism
last mentioned has sometimes been
used.
Fixing, lastly, the link c, we get the
less familiar mechanism shown in*Fig.
17 (C^P^V*. The link b swings about 3,
Fig.1673
and the crank a rotates in space somewhat
as in the meehanism (C"4) c, which we
have already seen This train has some
practical applications in machinery, but
is not very often nsed.
Here, then, we have obtained from
one and the same chain fonr entirely dif- *74
ferent mechanisms, ali of them more or
less familiar. The method we have
adopted has again been successful in
making the real relations of these appa-
rently dissimilar things perfectly obvi-
ous.
We have seen in connection with Fig.
14 that we can to a certain extent alter
the size and extent of a pair of elements
without altering its nature or changing
the motion of the chain to which it
belongs. This alteration in the size of
elements, or what may be called the
uexpansion” of elements, is a process
continually carried out by engineers for
practical constructive reasons; and often
gives to identical mechanisms extremely
different forms. It is impossible here to
go into this in detail, a somewhat ex-
treme case of it is shown, for the sake of
illustration, in Fig. 18. Here we have
the mechanism shown already in Fig. 8,
(C^P-l)4*. The pin of the pair 2 is so
enlarged as to include altogether the
pair 3, the connecting-rod b being simply
a circular disc, with an eccentric cylindric75
hole infit. The pin 1 again (the “ erank
shaft ”), is made large enough to include
the whole of 2. We liave therefore 3
within 2, and 2 within 1. We have one76
very common illustration of the extent
to which this expansion of pairs is car-
ried practically in the link motion. The
curved hnk and block are in their kine-
matie relations simply a very much ex-
panded pair of cylindric elements reduced
in extent by use of a process similar to
that by which we got Fig. 14 from Fig.
10.
We have seen what results have been
obtained by making two links of the
chain (C'J infinitely long. The same
process can be carried stili further. In
the familiar chain shown in Fig. 7, for
instance, three links, d, e and b, are made
infinite. We have therefore another
prism pair in it, and its formula becomes
(C^^-L,). It gives us the two mechan-
isms already mentioned and a third one,
ail of which are practically applied.
We must pass over without mention
many other modifications and alterations
of the chain, and mention only one
other form in which it oecurs, a form
which has some special interest. The
condition of movability of a chain that77
contains four cylinder pairs is not tbat
their axes should be parallel, but tliat
they shonld meet in one point. The

axes are parallel only in the special case
where this point is at an infinite dis-
tance. Fig. 19 is an illustration of the
more general, althougli less familiar, case78
when the point of intersection is at a
finite distance. This chain, which may
be indicated by the formula (CM* has
again its four inversions, and fnmishes
ns with three different mechanisms as in
the case of (C"4). I cannot here go into
these; I mention the chain partly be-
cause of its theoretic interest, and partly
becanse—although it looks so unfamiliar,
it is not unfrequently applied in ma-
chinery. If instead of snbtending only
a small angle, as in Fig. 19, three of the
links were made to subtend an angle of
90°, we should have the common Hooke’s
or universa! joint. In these “ conic ”
chains, links which are quadrants take
the place of the infinite links, so that a
chain having three quadrants corresponds
to the chain of Fig. 7, in which there are
three infinite links. The formula of the
universa! joint is (CJ- C L )°; it corresponds
exactly to (C"2 PJ-)a and is analogous to
(C"4)« and (C"3Px)« Figs. 11 and 16
respectively. I have already mentioned
in passing that the mechanism (C"sP9J-)a
* “ C four oblique.79
is that of 01dham’s coupling. We see
then clearly the close relationship in
which this coupling (for parallel shafts)
stands to the Hookes joint or coupling
for inclined shafts. It is another illus-
tration of the important results which
follow naturally from Reuleaux’s simple
method of analysis, and which hardly
seem attainable, certainly not with equal
directness, by any other method.
We may notice one other way in which
the chain (C"4), which we have seen in
so many forms, may be modified. In
some cases we may not wish to utilize
the motion of all the four links, but (say)
only three of them; in that case we may
omit the fourth link, if we carry out a
proper pairing between the two links
thus left unconnected. If, for instance,
we omit the link b we must make a
proper pairing between a and d. This
pairing will always be higher; the chain
becomes a reduced chain. Such a chain,
reduced by b, is shown in Fig. 20. The
higher pairing is carried out by placing
upon a a suitable element, here a circu-80
lar pin at 2, and giving to c tlie form of
the envelope of the motion of tlie
pin relatively to it. I liis envelope is
the curved slot shown in the fignre. It
was not necessary to take the new ele-
ment at 2, but if it had been in any
other place, as 2', the form of the slot81
would have been more complex, as is
shown by dotted lines. For such a chain
we may use the formula)—b. The
process of reduction can be carried on in
this way until only two links are left,
which then become really a pair of
higher elements. It is constantly em-
ployed in machinery, mostly in the case
of compound chains, or chains in which
some links contain more than two ele-
ments.
The chain which we have been examin-
ing has been applied more often than
any other to the leading trains of engines
and pumps. We shall in conclusion look
at a few of these machines, in order to
notice the constructive disguises which
often appear in them and render their
kinematic identity almost unrecognizable.
Fig. 21 shows a rotary engine which
has been patented several times, and
which is founded on the same* meehan-
ism, (C"sPJ*)d, as the common direct-act-
ing engine. The letters and tigures
placed upon it correspond to those on
Fig. 8, so that the identity of the me-82
chanisms may be the more easily traced.
The extraordinary change of form under-
Fig.21
gone by the connecting rod b is worth
special notice. It has become a bar
having a cross section like a half moon.83
It stili consists however, kinematically,
of two cylinders or portione of them, one
described about the axis of 2 (the center
of the dise), the other about the axis of
3. The element 2 is expanded so as to
include 1, the crank a becoming therefore
a disc. The mechanism is so propor-
tioned that the virtual length of the con-
necting rod—the dis Lance, that is, be-
tween the centers of the elements 2 and
3, is equal to the radius of the disc a.
With the mechanism in the form shown
in Fig. 21 some separate means has to
be provided for keeping b against a
when the latter is moving upwards.
Fig. 22 is another rotary engine, which
has been patented a dozen times since its
first invention in 1805. It is based upon
the mechanism of Fig. 16 ( C"% P*L)a.
The fixed link a is here made the steam
cylinder, while d becomes a moving pis-
ton. The reference letters are the same
as before. It will be noticed that the
cylindric element of the link c is expand-
ed to include its prismatic element; in
ali the cases formerly noticed the latter
had been the larger.84
In order to illustrate this part of his
subjeet Keauleux examines (in the work I
ha ve already mentioned) some forty or
fifty rotary engines and pumps ali de-
rived from the (C'\) chain and such
modifications of it as we have been look-
ing at. Models of a number of these are
now on the table before you. Many of85
them bear scarcely any extemal resem-
blance to the common steam engine, to
which they are, notwithstanding their
dissimilarity, so closely related. We are
not now concemed as to which form is
absolutely the best, but are only looking
at them from a kinematic point of view.
But it is worth notieing that in very
many cases not only constructive but
mechanical advantages have been claimed
for them. Their inventors have over and
over again claimed some mechanical gain
more or less mysterious, compared with
the ordinary form of engine. Most of
them, moreover, have been called “ro-
tary ” to distinguish them from recipro-
catui g engines. Our method of analysis,
although only kinematic, has shown us
not only that there can be no mechanical
advantage possessed by one over the
other, but that the word “ rotary ” is es-
sentially a misnomer, if it be supposed to
indicate that there is any more or differ-
ent rotation in them than in ordinary
engines.
We shall now only notice one more of86
these engines. It is one which has puz-
zled people a great deal, and with very
good reason, for its motionis are very
strange, and its analysis apparently very
complex.
The engine I refer to is shown (in one
form) in Fig. 23. It is known as the
“disc engine/’ It was brought a good
deal into notice in 1851, and for a few
years was used in the Times office witli-
out any ultimate success. Analysis shows
that it is based upon the same chain as
the Hooke’s joint, the conic chain, namely,
in which three out of four links subtend
right angles, and of which the formula
is (3 CL )a . The fixed link is, how-
ever, d, while in Hooke’s joint a (the
acute-angled link) is fixed. Patents
ha ve been taken out also for disc engines
in which a, and others in which 5, is the
fixed link.
I may say, in conclusion, that while it
may be impossible for many educational
institutione in this country to possess
themselves of models so perfect in exe-
cution, and therefore so expensive, asV
Fig.2388
those which have been sent to the Loan
Collection, it is yet quite practicable to
construet many of them in a form which,
while quite cheap, is yet well adopted for
educational purposes. I have some made
in this way of hard wood with brass
pine, which are very serviceable for coi
lege purposes. It is just the simpler
models, which are the most easily made,
which are the most useful in instruction.
I venture to hope that the treatment
of the theory of mechanism illustrated
by the models now in the South Ken-
sirigton Collection, and of which I have
here endeavored to set forth some lead-
ing principies, may prove valuable in
aiding the study and the comprehension
of this braneh of maehine Science both
to the student and the engineer.%* Any book in this Cataloguc scnt free by maU, Mi
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