ON QUADRATURES AND INTERPOLATION. 341 



Let Aa, Hh, Cc be throe consecutive equidistant ordiuates. Then, by 

 the polygonal rule, the area is represented by the trapeziums AahJi 

 and HbcC. Let at, ic be tangents at the extremities of the curve, and 

 draw a t' parallel to tc. Then the actual area of the curve regarded 



as a common parabola is the trapezium AacC plii,s |^ of the triangle 



o 



ate (= t at') while the polygonal area Aa h c C is AacC+-~atc 



= A ac 



C + - t at', and the difference between these is ( ^ — — j tat' 



= \tai'. 



D 



When more oi'dinates are used, it is easy, by repeating the construciion, 

 to form a triangle which shall give a superior limit to the error made by 

 substituting the trapezoidal rule for the parabolic. For the geometrical 

 addition of the curvilinear segments, taking each as two-thirds of its 

 circumscribing triangle corresponds very nearly (although not exactly) 

 'to an algebraical addition which can be effected graphically on the second 

 ordinate from each end, by drawing parallels to the chords and tangents 

 from the head of the first ordinates.* It also follows that if the tangents 

 at the extremities of the curve are para;llel, the difference between the two 

 rules disappears, and they lead to a result practically identical — that is to 

 say, only differing by an error of a high order. 



Section 4. — MuUi/de integrals, ordinates not differenced. 



A multiple integral of the form / / xh dx dy, in which 



the limits ai'e all constant, and the variables (except m) all independent, 

 can be computed by treating each variable separately, by a repeated appli- 

 cation of any process of arithmetical integration. In the case of a double 

 integral applied to the calculation of volume this is equivalent to cutting 

 the volume by parallel plane sections, obtaining the areas of these by any 

 method of ordinates, and then summing the areas of these sections, each 

 taken as an ordinate, to obtain the volume. Thus, taking nine equidistant 

 ordinates with the interval h in one direction, and Tc in another at right 

 angles to it, and calling them 



tti &1 Ci 

 «2 ^2 ^2 

 O'.i ^3 C3 



an application of Simpson's rule gives us for the plane areas 



—h («! + 46i + Ci),- h (rtj + 4&2 + C2), ^ h {a^ + 4Z>3 + C3) 



or, using the vertical sets, 



I Ic (a, + 4*2 + a^),^lc (h, + U^ + h^), ^ J: (c, + 4c., +C3) 



* See Woolhowse 'On Interpolation, Ac' Assurance Magazine, vol. xi. p. 308 — sepa- 

 rately published by C. and E. Layton in 1865. See also Leclert, 'Note sur le Calcul 

 numerique des aires curvilignes planes,' Annates du Genie civil, tome viii. p. 630. 

 M. Leclert states that his note is in great part a reproduction of M. fifiech's lessons. 



