M E N
179
ME N
th,
before £ , „2r . re 3 + GC
ni
T X
tr a
^ m
AE= _X
26 — ,
X 2
(2 b~a-
Pezoid abbe
c) the area bcdb; but the area of the
(2 6 .
— equal,therefore the area
■c) X
2 m
~3~
f X (a + c) X 2 m = (a -f 6)
ABCBEA “ (a -j- <?) X »» +
/ 46 — 2 a — 2c\
= fa + c H---J X rn
+ 3 C + 46 — 2a
2 c
x i» =
a + c + 4 6
m,
> or
a + 4 6 + c
X m.
Ki
h j r ^ et A b c d e p g, &c. be the abscissa of a curve,eq u ; ( j. r ' SI N > &c. and let A h, b i, C j, be any number of even«, j lst ant ordinates, and let these ordinates be denoted byPass tl ! a ’ e > f> 9> &c. Let us suppose a parabolic curve tot\ Vo every adjoining three ; that is, taking in every
«v e ,. v l°' n ing spaces ; and let m be the common breadth ofs pace ; then the area of the first two spaces will be
2+46 + V
m ; the area of the two next spaces will bem; the area of the two next spaces will
X m, and so on, as far as we please. Now
these areas will be the area of the whole> therefore a + 46 + cHa.. ’ a
8,1,11 of a nClJ <v e
c + 4 d + e
m 4--- X m, +
X m
X m & c —— + 4 ^ + 4/+ 2c + 2e + a + g’ ' ~ 3
X Jb + d + /) + 2 x (c + e) + a + 9
g
Th‘ ° aiea tbe curve-
^ ’l'h er , n j' 1 on ly holds true in superficies, but also in solids.e hUto^p ! be intermediate ordinates be called sections, and'| et| ’ an(]' e<i 2> &c, the first being called odd, the next
t ■ » then 80 ° n a 'ternately ; the last section will always be' Vice tl.» _ to tour times
odd': so .^
-j the* 1 to * our times the sum of the odd sections addI'O'thirq °f the even sections, and the two ends ; thent|, e the sum being multiplied by the common distancefNr fin.„ area > which is a near approximation for any curvi-whatever -
ions on Averaging Surfaces, bounded on one■ o' th e R °N two opposite sides bt a Curve .
° n ^othoj^^tionof superficies bounded by i
c curves, the com-
Ot^i othod of t
by tes, ,akln g the average, or mean, of any number of
i* ' e 'r numb ad t'' e ordinates together, and dividing
j^terrieiy v ° r ’ and multiplying the quotient by the length,or a 2U p
i"*
bi
7 figure proposed to be measured be A _ A1Csnot \ tbe equidistant ordinates B V c ’ and
l> d b y «, B i by 6, ch by c.vh by d * J ^ m .
"‘let ea'2 mon distance > AB, or B C, &c., DEFGD,
1 <w ach of the areas, abika, bchib, CD® « c, de^ According to the method of measunnga trap ^
^ s en the sum of the contents will be K t xH ho,” he bac *> A E F H k a, provided that the lines, ' whenai ®ht, and very nearly equal to the . exces8
e ls bounded on one side by a curve; being
c
‘he
when kiiigf is concave, and in defect when convex:then will
a + 6
XlBrABIKA,
and
2
b + c2
c + d2
d + e
X m = BCHIB,XM = CDGHC,
X fflr DEFGD.
The sum of these areas is + 6 + c + <fJ»i; that
is, if the half sum of the extreme ordinates be added to the tintermediate ones, and the sum multiplied by the commondistance, the product will give the area : and it is evidentthat this will always be the case, whatever be the number ofordinates.
Therefore, let a, 6, c, d, &c., to p and y, be any series ofordinates whatever, whereof a is the first term, and p and qthe two last; then the area of the curvo-rectilinear space will
be generally expressed by (— ~ q + 6 + c + d, &c. to pj
X m ; now, let l = the length ; then if n be equal to
the number of ordinates, n — 1 will be the number of
spaces, or areas; therefore ^ + 6 + c + d, &c. to pj
— + 6 + c + d, &c. to p) ,
X m = { 2 x l —
n — 1
a + » + 26 + 2c + 2c?, &c. to 2 p
" ' —f and the value of
2 n — 2
. a + 6 + c + <?, &c. to p + q
the average method, viz. - - --
n
. a + 6 + c + <?, &c. to p + q
X m X (n — 1) = —^^X l;
71
then putting one of these equal to the other, there will result
n x ^ = a+ 6 + c + c?, &c. to p + q.
It appears, therefore, that there can be no equality exceptwhen the half sum of the extreme ordinates, multiplied bythe whole number of terms, is equal to the sum of all theordinates ; which, therefore, must be in arithmetical progres-sion, or amount to such.
The method of averaging will therefore be very uncertain ;as the half sum of the two extreme terms, multiplied by thenumber of ordinates, can be taken in any ratio, with respectto the sum of the ordinates.
The difference between the average method and the truemethod, would be the very same as the difference betweena + 9 , 7 a + b + c + d, kc. top + q
2 X (n —1) n x (n —1) ’
for if from two unequal quantities equal quantities be takenaway, their difference will still be the same; and if to twounequal quantities equal quantities be added, the differencebetween the sums will still be the same.
If equimultiples be taken of unequal quantities, the differ-ence between the products will be the same multiple of thedifference ; and if any aliquot part, or equisubmultiple, betaken of two unequal quantities, the difference will be equal