DOM
286
DOM
i i
Let s equal the side of the square base,
£
then — is equal to half the side of the base ;
and if d be the diameter of the section, and a its altitude,then, by the property of the circle,
t- — a (d — a) — da — a 24
« 2
therefore, do = - + a>s 2
consequently, d = — + a4 a
and 2 d z= £_ + 2 a2a$2
Therefore, by substituting — + 2 a for 2 d in the formula4a 3 as 2 2a 3
2 a a 2 -g-, we obtain — + -g- for the solidity of the
segmental dome, independent of the diameter of the section.This rule may be expressed thus :
To half the area of the square base, multiplied into thealtitude, add two-thirds of the cube of the altitude, and thesum will be the solidity of the dome.
Let us take the same example as at first, and we shall findthe side of the square of the base to be 16 feet: therefore,16 4
16 4
96
16
16
4
256 square of the base. 64 cube of the altitude.4 altitude. 2
2)1024
512
42f
3)128
42§
554§ cubic feet, as before, in the segmental dome.
The solidity of any dome whatever may be found by thefollowing general rule :
To the areas of the two ends, add four times the area inthe middle ; then, one-sixth of the sum multiplied by thealtitude, gives the solid contents. This rule applies to alli i domes whose vertical section is contained between any twoopposite arcs of the same circle, and two parallel lines, andwill even apply to those domes which are the segment of asphere. In the second example, the side of the square ofthe base being 20, and of the top 16, the middle area will befound to be 364.
20
20
400 area of base.
16
16
96
16
256 area of top.
364
4
1456
256
400
2112 solidity.
Because multiplying and dividing by 6, gives the samenumber.
To find the solidity of a hollow square truncated dome, theshell being of equal thickness, supposing each edge of the baseequal to the diatneter of the circle of which the section isa part.
We found before, supposing d =: the side of the square
-" Tthe s° lii
base, and a equal the altitude, that the solidity o
4 a s , the gol^
dome was expressed by <P a -, then to fine 1 ^
^ , two ^ »
of the shell, it is only finding the solidity ot s ; 0 ij 9 411domes, of the same altitude, but of different dimthe base, and deducting the greater from the less- sa ifs ce>Now let d be the side of the base to the externa^ gut'and d the side of the base corresponding to the ‘ n -(bin
face ; then the solidity of the solid comprehended - D g tb e
external surface, and the two parallel planes I
4a 3
ends, is d 2 b —•
In like manner, the solidity oflthe solid which would fill!the cavity, is J
then, by subtracting the latter I
d 2 a —
3
4b 3 ,
-* J
3
.»)
en, oy subtracting tne muur i , •
of these expressions from > d 2 a — d 2 a ^ a \the former, we obtain Jfor the solidity of the shell.
This rule may be thus expressed in words : by $
Multiply the difference of the areas of the g0 lidi;altitude of the dome, and the product will g iveof the shell. hcW eeI> \e
Example .—Suppose the side of the base, 0 {external convex surface, to be 20 feet, and the t
base of the internal cavity, or bason, to besolidity of the shell is required.
18
18
144
18
324
20
20
the c0 °
400 area of the base contained between c0 ncS324 area of the base contained betweensurface.
76 difference of the areas of the bases*
6
456 solidity of the shell, as require •
To find the convex surface-of a dome.
Let the diameter b g = d
BA = *
and c a —yb c = z
we have, by similar triangles*
A o c and c e d, c a : c o :: o E : 0 .
d z. ■ e ”
that is, y : -
2^ „ A{C r
but since the fluxion of the surface whose .se fi >lar is denoted by 2 pyz, where p is equal o ^
therefore, we have 2 pyz — P~, X . l °i
* ■ •, eop , g t
■
are
pd x=; P£ tbe se »t becomes a hemisphere, *
p& 2 Slnee P = 4 x .7854, we shall b* rt>
spheric dom S ^ ^ the convex area of the
h 0f legmen tald be ^ ° f its base; a °f
be .same altitude „ 13 the same as that of a of^
g eat circle ; it f 0 ]i n ’ ^of a diameter equal to & <• $e?Vacated dome i s ~ ? SOj that tbe convex surface 0 ^ ( m
^altitude, and nf^J 0 the surface of a cyhoder^
tbe sphere. d of 3 diameter equal to the great c**