ON QUADKAIUKES AND INTERPOLATION. 337 



This is a modification of the rule of seven ordinates already given, 



differinj? from it only by — — A^.* 

 ° •' -^ 140 



It will be observed that these formulas are symmetrical end for end, 



and since the integration is between definite limits, the origin of the 



abscissa; is indeterminate, and may be taken so as to fall in the middle, or 



so that the equivalent integration is from — rh to + rlo. It follows that, 



for the purposes of comparison, instead of taking 



1/ =?)(, + 6j jB + &2 ^^ + ......... &„a;" 



we may take, since 



2.2m- 1 ^jj. __ Q always 



/: 



1 



,2 



y = K + l.2x'' + t,„on;2'« 



omitting the terms containing odd powers of x; and that we therefore 

 obtain no greater generality by using 2 m ordinates instead of 2 m — 1. 

 The error in either way is of the same order. f 



The formula; of quadrature for an odd number of ordinates or an even 

 number of intervals appear to have been also given by James Stirling,;!: 

 who adds a set of what he calls corrections. The number of ordinates 

 being 1in — 1, the correction is of the foi-m 



- AE-"'(E - 1)2'" X base 



the coefficient A being apparently determined so as to make the corrected 

 value exactly agree with the result obtained from integration, when both 

 are applied to a;^'". But they do not lead to the next rule, for 2 m + 1 

 ordinates. As a particular example, the correction for the rule of three 



ordinates, namely, -,- (^^-i + 4ifo + ih') x base, is — ---. (w_2 — 4m_i 

 b 180 



+ 6mo — 4mi + u-i) X base. The values of the coefficients A are inexactly 



given by Stirling as 



1111 



instead of 



180 ' 470 ' 930 ' 1600 

 12 3 296 



180 ' 945 ' 2800 ' 467775 



The inaccuracy is not a mistake, because Stirling only uses them as a 

 test of approximation, and not as a means of obtaining accuracy. 



Bertrand (Cahul Integral) gives the corrections in a slightly difierent 

 form, from which the coefficients just given are obtainable by multiplying 

 Bertrand's corresponding coefficient by 2^'" : A^"' 0^'". In the following 

 table we give Bertrand's first coefficient only. It is the excess of the com- 



* The first eight formula; are given by Tliomas Simpson and verified by Atwood. 

 All the rules are given (but with some misprints) by Bertrand {Calcul Inti'gml). 

 Atwood makes a curious mistake in the rule of 8 ordinates. He is endeavouring 

 to correct Simpson, with whom, however, his result is really identical, only that 

 Atwood has introduced the factor 49 into both numerator and determinator, without 

 seeing that it divides out : see A Disquisition on, the StaMlitij of Shij)s, by George 

 Atwood, F.R.S., read before the Royal Society, March 8, 1798, and reprinted 

 separately (p. G2 of reprint). 



t See Todhunter, Onthi' Functions of Laplace, 3,'c. pp. 98 and 104. 



j See his Methodus Differentialis, London, 1730, p. 14G, Prop. xxxi. He stops at 

 nine ordinates. 



1880. z 



