Encyclopaedia Britannica, or, a Dictionary of arts, sciences, and miscellaneous literature : enlarged and improved. Illustrated with nearly six hundred engravings > Volume 13, MAT-MIC

M E C H
Theory. This is obvious from the laft paragraph, for the pieces
of chain a m, b n, c o, k\J, &c. are forces afting upon
the points a, b, c, k of the catenary, and are proportional
to a m, b n, c 0, &c. the distances of the points a, b, ct
i, &c. from the roadway.
322. An arch of this conftruftion will evidently an-
fwer for a bridge, in which the weight of the materials
between the roadway and the arch ftones is to the
weight of the arch ftones, as the weight of all the pieces
of chain fufpended from a, b, c, &c. is to the weight of
the chain A £ B. As the ratio, howrever, of the weight
of the arch ftones to the weight of the fuperincumbent
materials is not known, we may affume a convenient
thicknefs for the arch ftones, and if from this aflumed
thicknefs their weight be computed, and be found to
have the required ratio to the weight of the incumbent
mafs, the curve already found will be a proper form
for the arch. But if the ratio is different from that of
the weight of the whole chain to the -weight of the fuf¬
pended chains j it may be eafily computed how much
muft be added to or fubtra&ed from the pieces of chain,
in order to make the ratios equal. The new curve
which the catenary then affumes, in confequence of the
change upon the length of the fufpended chains, will
be the form of an equilibrated arch, the weight of whofe
arch ftones is equal to that which we affumed.
Scholium.
323. In moft cafes the catenarian curve thus deter¬
mined will approach very near to a circular arc equal
to 120 degrees, which fprings from the piers fo as to
form an angle of 60 degrees with the horizon. The
form of the arch, however, as determined in the pre¬
ceding propofition, is fuited only to thofe cafes in which
the fuperincumbent materials exert^a vertical preffure.
A quantity of loofe earth and gravel exerts a preffure
in almoft every diredtion, and therefore tends to deftroy
the equilibrium of a catenarian arch. This tendency,
however, may be removed by giving the arch a greater
curvature towards the piers. This will make it approach
to the form of an ellipfis, and make it fpring more ver¬
tically from the piers or abutments.
324. We ftiall now proceed to deduce the form of an
arch and its roadway, by eftabliftiing an equilibrium a-
mong the weights of all the materials between the arch
and the roadway. Ihis method was given by Emerfon
in his Fluxions, publifhed in 1742, and afterwards by
Dr Hutton in his excellent work on bridges.
Prop. II.
325. To determine the form of the roadway or
extrados, when the form of the arch or intrados
is given.
g. 8- Let the lines AD, DE, EB, BF, FG, GH lie in
the fame plane, and let them be placed perpendicular
to the horizon. From the points D, E, B, &c. draAV the
vertical lines D d, Ee, B b, &c. and taking D p of any
length, make E r equal to D p, &c. and complete the
parallelograms pc,qr. Again, make B szzqe, and com¬
plete the parallelogram /r; in like manner make Ek^isb,
and complete the parallelogram F f ; and fo on with all
the other lines, making the fide of each parallelogram
©qual to that fide of the preceding parallelogram which
A N I C S.
is parallel to it. Let us new flippofe that the lines
CD, DE, EB, &c. can move round the angular points
D, E, B, F, &c. the extremities A, C being immove¬
able ; and that forces proportional to Dr/, Ee, BZ», &c.
are exerted upon the points D, E, B, F, &c. and in
the direflion Dr/, Ee, &c. Now, by the refolution of
forces, the force Dr/ may be refolved into the forces
D c, D p, the force E e into the forces E y, E r, and the
force B b into the forces B j-, B /, and fo on -with the
reft. The force D c produces no other effedt than to
prefs the point A on the plane on which it refts, and is
therefore deftroyed by the refiftance of that plane 5 but
the remaining force Dp tends to bring the point D to.
wards E, and to enlarge the angle ADE 5 this force,
however, is deftroyed by the equal and oppofite force
E q, and in the fame way the forces E r, B t, F x are
deftroyed by the equal and oppofite forces Br, Ek, G v,
while the remaining force G u> is deftroyed by the re¬
fiftance of the plane which fupports the point C. When
the lines AD, DE, &c. therefore are adted upon by
vertical forces proportional to D </, E e, B b, &c. thefe
forces are all deftroyed by equal and oppofite ones, and
the lines will remain in equilibrio.
326. Now the force Dc : Dja or Eyrzfin. cdT> or
dE)p \ fin. AD </, that is, by taking the reciprocals
D r : E
* 4-,
and for the fame reafon
fin. AD d' fin. </ D jp’
E y : B szz
Hence
fin. E eq ‘ fin. b B s‘
E?~fin. Ee^
Now, fince Eq : Ee=fin. Ee q : fin. E qe, we have
EyXfin. Eqe .
Ee=—-— that is, fince DE/w — E qe. and
im. Eeq 7 ’
t"n t- E <7 x fin. DEm ^ ^
.EB=Eef; E,= -L* s ButE?=;
fin. e EB
therefore, by fubftitution, we obtain
"fin. Eeq
£ ^ fin. DEm
’"r" fin. EeqX fin. e EB "
Now, as the fame reafoning may be employed to find
D c?, B b, &c. we have obtained expreflions of the
forces which, when adting at the angular points D,
E, B, &c. keep the whole in equilibrio, and thefe ex¬
preflions are in terms of the angles which the lines DE,
EB, &c. form with the diredtion of the forces. If the
lines AD, DE, &c. be increafed in number fo that
they may form a polygon with an infinite number of
fides, -which will not differ from a curve line, then the
forces will adt at every point of the curve, and the lin*
m E will be a tangent to the curve at the point E, and
DE m will be the angle of contadl. The line E q be¬
ing now infinitely fmall will coincide with E m, and
therefore the angles eEq and eEB or Eeq will be
equal to the angle e E m, and confequently their fines
will be equal. Therefore by making thefe fubftitutions
in the laft formula, we have an expreflion of the force
at every point of the curve, thus
fin. DEm
l fin. DE m
fin. E w * fin. r E w £
But