A = , udx = C1 + [124h2u′ + 1720h4u′′′ − 130240h6uv + 11209600 − . . .]𝑥＝𝑥𝑚
𝑥＝𝑥0;
and an analogous formula (which may be obtained by substituting 12ℎ and C12 for ℎ and C1 in the above and then expressing T1 as 2C12 −C1)
A= 3f;"udx=T +, § h'u'-rfm, h'u”'+w$-§ f6h°uvf n, g § %8 Whsuvi1+
To apply these, the differential coefficients have to be expressed in terms of differences.
76. If we know not only the ordinates u0, u1, . . . or ui, ug, , but also a sufficient number of the ordinates obtained by continuing the series outside the trapezette, at both extremities, we can use central-difference formulae, which are by far the most convenient. The formulae of § 75 give
A = C1 477' ' i12l"5u'l"712h5 l45'“.'rf5ifzs#55“ +ra'2*a°s'mrH5'“ " ~ - - x =x" x=x0
A=T»+h aaa- saeu+f¢° ° 1 »°u-aasiswiiu+ .. 3253"
77. If we do not know values of u outside the figure, we must use advancing or receding differences. The formulae usually employed are
A=C1+h {§ Au0-,1;A'u0+-,1, i6A3u°-, § UA4u0+ . +i1, A'u, ,, -,1;A'1'u, ,, +»,1§ , ;A'3u, ,, —f~2¢, A"u, ,, +
A ='r, +h{ -,1, Au, +,1, A=u; -, w, , a=u¢+,1, °, #, ,A4u;- —{;A'14m-{+ s11A"u -§ - =*»f"n%A'”u»..; + s't°§ %A"'u»
where ∆, ∆2, have the usual meaning (∆u0=u1-un, A'u0= Au; - Aug, . .), and A', A”, denote differences read backwards, so that A'u, ,.= u, ,, 1 -u, ,, , A"u, ,, =u, ,, 2- 2u, n,1-i-ilm, . . The calculation of the expressions in brackets may be simplified by taking the pairs in terms from the outside; 'i.e by finding the successive differences of ug -|- u, ,, , 'ui + um-1, , or of ui + u, ,, , *, 1¢§ 'i'Um-gi - —
An alternative method, which is in some ways preferable, is to complete the table of differences by repeating the differences of 'the highest order that will be taken into account (see Interpolation), and then to use central-difference formulae.
78. In order to find the corrections in respect of the terms shown in square brackets in the formulae of § 75, certain ordinates other than those used for Ci or T1 are sometimes found specially. Parmenliefs rule, for instance, assumes that in addition to up 'ui . . u, ..,4, we know no and um; and u*-ue and u, ,, -u, ,, § are taken to be equal to #huh and %hu', ,, respectively. These methods are not to be recommended except in s ecial cases.
79. By replacing ℎ in § 75 by 2ℎ, 3ℎ, . . . and eliminating ℎ2u', h'u"', ., we obtain exact formulae corresponding to the proximate formulae of § 70. The following are the results (for tffe formulae involving chordal areas), given in terms of differential coefficients and of central differences. They are not so convenient as the formulae of § 76, but they serve to indicate the degree of accuracy of the approximate formulae. The expressions in square brackets are in each case to be taken as relating to the extreme values x=x0 and x=x, ,., as in §§ 75 and 76.
(i) A=§ (4C1-Cz)+[-;§ »5h'u"'+-f§ fgh°u'f-T;}1;5h°uvil+ ] =t(4C1'C2)+ℎ[−1180u+rs"'r1rl15'1¢-ro°v°r'1r6#5'14+. . .].
(ii) A=%(9C1-Ca)+[-s*rh'“"'+ rhh°“Y'rr1¢“rsh“1¢"“+ - - -l =i(9Ci-Cs)+h[-n'iw5'u+t?ir#5°1¢-rn"*s°&roM5'14+- - -]
(iii) A=, 1g(64C1- 20C2+ Cr) +[-;»{, h°u"+;;-2, ~¢;h”u”“- ] =I1s(@4C1'“20C2+C4}+hl-9iirP»55U 'l'tiii%rM5"1¢- ~ - - l
(iv) A={~, ,(l5C1-6Cg+C3)+[-§ }5h°u" 1-;;H, -;;h“u"“- ] =#a(I5Ci-6Cz+ C=)-l-hi- tiUM5°1¢'i'sis"x'v6M5714- » - - ]
(v) A=|1¢(56C1-28Cz-l-SC;-C4)'l'l ~{1Hr5/z8u"“-Q' . ] =9l;(56Ci-28C¢-I-8C;—C4)+h[—2-;1n;tE7u+
The general expression, if p, q, r, . are k of the factors of m, is A=PC, +QC, + RCf+ . + ()'=b, h='=§§ f,2 + - d“'°+1u x = x
k+1 z1=+z . fn
() b'”'1h dx2"+1+°" x=xu
where P, Q, R, have the values given by the equations in § 71, and the coefficients bk, b;, +1, are found from the corresponding coefficients in the 'Euler-Maclaurin formula (§ 75) by multiplying them by Pp'-"+Qq”'+Rr”°+ ., Pp2"*'+Qq”'*'+I{r”'*“2+. ,
80. Moments of a Trapezette.-The above methods can be applied, as in §§ 59 and 60, to finding the moments of a trapezette, when the data are a series of ordinates. To find the pth moment, when ug, ul, u¢, are given, we have only to find the area of a trapezette whose ordinates are xgpug, xipul, xgpuz,
81. There is, however, a certain set of cases, occurring in statistics, in which the data are not a series of ordinates, but the areas AQ, Ap . Am-; of the strips -bounded by the consecutive ordinates ug, ul, . . um. The determination of the moments in these cases involves special methods, which are considered in the next two sections, »
82. The most simple case is that in which the trapezette tapers out in such a way that the curve forming its top has very close contact, at its extremities, with the base; in other words, the differential coefficients u', u", 'u”', are practically negligible for x=x0 and for x=x, ,, . The method adopted in these cases is to treat the areas Ar Ai, . . as if they were ordinates placed at the points for which x=x*, x=x§ , , to calculate the moments on this assumption, and then to apply certain corrections. If the first, second, moments, so calculated, before correction are denoted by ρ1, ρ2, , we have
pi = x;A;+ xgA; -- .' + x, ,, ;A, ,, ; me x'iA=i + x'eAs + . . - + x', . iA ..§ p, ,=xPiA§ -I-xPgAg+ +xP, ,, ;A, ,, ;-
denoted by vi, va, - .
These are called the raw moments. Then, if the true moments are their values are given by
ν1≃ ρ1
V2 £132 '- 32 hfpo
Va¥Pa '” U12/Ji
ν4≃ ρ4 −h2ρ2 +7240h4ρ0
1's£Ps - ihzpa +;ZsVh"P1
where ρ0 (or ν0) is the total area A; -|- A; + . + A, ,, ;; the general expression being
!
1"p£Pp", H 2 !|(p 2 lhzpp-2 "l" M 4! P Zn h'Pp-4- . where ~
>~i=1'r» }'2=?1'6v }s=r§ izn 7=a's%»?s=a”r"a“t"r, - - » The establishment of these formulae involves the use of the integral ca cu us. 1
The position of the central ordinate is given by x=v1/po, and therefore is given approximately by x-pl/po. To find the moments with regard to the central ordinate, we must use this approximate value, and transform by means of the formulae given in § ~32 This can be done either before or after the above corrections are made. If the transformation is made first, and if the resulting raw moments with regard to the (approximate) central ordinate are o,1r2,1r3, . . ., the true moments ui, 112, pa, with regard to the central ordinate are given by
H1 = 0
- flirt-1'r7l'po
/-¢a&'Ifs
u4& vu °-' ihirz +512-6 h*/Ja
1/»r%1fa"~§ h21fs
83. These results may be extended to the calculation of an expression of the form f;;"u¢(x)dx, where ¢(x) is a definite function of ~x, and the conditions with regard to u are the same as in § 82. (i) If ¢>(x) is an explicit function of x, we have f;;"'u¢(x)dx4-@Ai¢(xt)+A¥//(x&)+ - + Am-ilf(x, ..i) where ., o(x)§ ¢(x)-§ h2¢"<x>+§§ h4¢fv(x>- ,
the coefficients M, M, ... having the values given in § 82. (ii) If 4>(x) is not given explicitly, but is tabulated for the values xy xg, . of x, the formula of (i) applies, provided We take i0(x)E(1-'s'x5'+r%r5'-n'r<:5“+ - - -)¢(x)-The formulae can be adapted to the case in which ¢(x) is tabulated for x=x0, xl, f -,
84. In cases other than those described in § 82, the pth moment with regard to the axis of u is given by
VP = x"mA -Psp-1,
where A is the total area of the original trapezette, and S, , 1 is the area of a trapezette whose ordinates at successive distances h, beginning and ending with the bounding ordinates, are . 0. x="“A;, x2'“"(Ai+~Ai), . - ~ x£ il(As+ AH- - - - + Am-i)» x5"A» The value of S, ,, has to be found by a quadrature-formula. The generalized formula is
f§§ 3'“¢»<x>dx = A¢<x > - T. T
