^ whole mass, and also the weight of the vaulting*dj °4 e join t H i. Therefore we have this proportion :J V + yi . (fy s /i , + y J = 6 e : bF = be : cg =

C G .

<3g. Therefore cg =

d y x J x 2 + y 2sdy J x 2 + f

*' If 1.

hi eq u -i'f fUrv ature of the dome be precisely such as puts itt er hc a i • . um > b ut without any mutual pressure in the*> tli e J 01 I 1 t3, this value of C G must be equal to C b, or toP°mt g coinciding with b. This condition will be

ex PresseJ k v ,v . dyx*/a? + f

u D y the equation —- — x or more con-

» d y i? 4 - y

’«ntly by dy V x l + if X_

, __= —. But this form gives only

Motte r - sd V-d ^ + y 2 x°'° ts a e( l u IIiI >r i urn J independent of the friction of thePjjnieq k 11 cohesion of the cement. An eqilibrium, accom-" l he v ^ s . 0me fi rm stability produced by the mutual pressureve rtical ‘ ’ - - -

of

joints, may be expressed by the formula

stf,r>=s±-.£L *, . dy n/ & +y 3 _x , t ,

» 7r + ‘

peases aila ^ e positive quantity which increases when x^' This last equation will also express the equil-i-* . °tt>e, if t be a constant quantity, because in this case

Si

•nee n n ,... . , d y x J ic 3 + f

, a firm stability requires that —^

Jall be 3 4 sdyj x 1 + y 2

*r e lce T f Fe 1 ater than x, and C G must be greater tjian C B.

■Its (j e team that figures of too great curvatures, whoset en< t too rapidly, are improper. Also since stability

that ,, , d y x «/ X 2 + y 2 ,

at we have ———-— greater than

Hot we learn that the upper part of the dome

"W “c niade very heavy. This, by diminishing theill ttie p 0 j °‘ ^ p to b c, diminishes the angle cJ G, and mayOr ^at v,/ 11 G above B, which will infallibly spring the dome0 j.P r 0s Se8 ace - We see here also, that the algebraic analysisfile vm lat peculiarity of dome-vaulting, that the weight

‘The

Part may even be suppressed.

ne fluent .v, • dy^x'+y 2 £ t’.

cut ot tbe e q ua t 10n ■ -rj —==- — -——is most! 31y fn , *dyj x 2 + y 2 x t

1 .Iperboii i 1S . T - * dy J x 2 + y 2 = l x + t-, where l is*y c logarithm of the quantity annexed to it. If wel ‘iag at aa constant and correct the fluent, so as to make it. Ie vertex, it may be expressed thus :

—t« = L*— l y + Lt. This gives

I "^^1 + y 2 x , , Al ^ * d y sf & + f

t . a- - — t y and therefore- - -

,: n * y <t

M

last P

0r Ration will easily give us the depth of thek * tiiickr ~ • ■

v

ness, d, of the arch, when the curve is

°r if.

Us fluxion is * • -+*■ = C54^~ and d

: 4 . a y

** + JT"’ vvhich is all expressed in known quantities:

4 o

for we may put in place of t any power or function of x or ofy, and thus convert the expression into another, which willstill be applicable to all sorts of curves.

x i

“ Instead of the second member r 4- -we might employ

x t

p x

where p is some number greater than unity. This will

evidently give a dome having stability ; because the originaldyx^/x 2 +f

formula — ../ ■ will be greater than ie. This will

sdy s /x 2 + ij 2 6

_p ax p — 1 xgive d~~

• Each of these forms has its advantagesEach of them also gives

yy p V x 2 +f

when applied to particular cases.

ax • .

d — - , when the curvature is such as in precise

yy V x ‘ + f

equilibrium : and lastly, if d be constant, that is, if the vault-ing be of uniform thickness, we obtain the form of the curve,because then the relation of;? to a; and toy is given.

“ The chief use of this analysis is to discover what curvesare improper for domes, or what portions of given curves maybe employed with safety.

“ The chief difficulty in the case of this analysis arises fromthe necessity of expressing the weight of the incumbent part,or sdy*/x 2 + f. This requires the measurement of theconoidal surface, which in most cases can be had only by-approximation, by means of infinite series. We cannotexpect that the generality of practical builders are familiarwith this branch of mathematics, and therefore will notengage in it here ; but content ourselves with giving suchinstances as can be understood by such as have thatmoderate mathematical knowledge, which every man shouldpossess, who takes the name of engineer.

“ The surface of any circular portion of a sphere isvery easily had, being equal to the circle inscribed with aradius equal to the arch. This radius is evidently equal to

V x 2 + f~.

“ In order to discover what portion of a hemisphere maybe employed (for it is evident we cannot employ the whole)when the thickness of the vaulting is uniform, we may recur

to the equation or formula x —sdy*J x‘ + y 2

Let a be the radius of the hemisphere. We havegyy ^ a f , n

* = ^ a i 71 yt and x = " j _ Substituting these values

in the formula, we obtain the equation y* */a? —y J =

g'yy

S .j d 1 ~~f We easily obtain the fluent of the second

member =: a 3 — a? a 1 — y 1 , and y = a */ —^

Therefore, if the radius of the hemisphere be one-half, the

breadth of the dome must not exceed — £ + ■/ or 0.786,and the height will be 618. The arch from the vertex isabout 51° 49', much more of the hemisphere cannot standeven, though aided by the cement, and by the friction of thecoursing joints. This last circumstance, by giving connectionto the upper parts, causes the whole to press more verticallyon the course below, and this diminishes the outward thrust jbut at the same time diminishes the mutual abutment of thevertical joints, which is a great cause of firmness in thevaulting. A Gothic dome, of which the upper part is aportion of a sphere not exceeding 45° from the vertex, and

t