ON QUADEATURKS AND INTERrOLATIOX. 371 



double integration is easily effected between the required limits, either by 

 WooUey's rule, or by any other method which may suit the case.* 



It is of some consequence to conduct the work so that the degree of 

 accuracy may be as nearly as possible the same throughout. It is best to 

 follow the rule of using as nearly as possible the same number of signifi- 

 cant figures right through. Thus it is idle to use five or six figures for a 

 moment, and only two or three for the coordinate of the centre of gravity, 

 because either there are more than we want for the moment, or else there 

 are not enough if we require any further step to be taken from the value 

 so found for the centre of gravity. It is a common thing to see much 

 good work disfigured by want of attention to this. The rule above sug- 

 gested is not absolute ; but it is on^he whole the best to work from, except 

 in some special cases, which a little thought will easily discriminate. 



It is scai'cely necessary to enter into any detailed disquisition con- 

 cerning the application of arithmetic to interpolations. Probably too 

 much has already been written on the subject. With regard to inter- 

 polation in two dimensions, the reader may usefully consult Legendre, 

 Fonctions ElUptiqaes, vol. ii. cap. xv. pp. 201-207. 



IX. — Graphical MExnocs. 



Of mere interpolation, there is no need to say anything here. When 

 once a function is represented by the ordinate of a curve, the interpolatiou 

 is effected at sight, whether the direct interpolation from the intermediate 

 value of the variable, or the inverse operation of obtaining the value of 

 the variable corresponding to a given value of the ordinate. "In the same 

 way a parallel ruler will give us the means of interpolating direction, and- 

 of finding maxima and minima. 



In the case of quadra,ture there is something to be said, although that 

 is very little more than the translation of the arithmetic into geometry. 

 The geometry is practically restricted to the simple parabolic rule, or to 

 the trapezoidal rule — the parabolas of higher order are of course tmsuited 

 to graphical work. That they are so has already been showm to be a 

 matter of no great consequence. 



The fundamental opeiation of quadrature is that of finding the aret 

 of a plane curve. It is convenient to reduce the construction to that of 

 finding the area included between a base line, two parallel ordinates at 

 i-ight angles to the base, and a curved line forming a fourth side to the 

 figure. As has been already remarked, this curve must never be parallel 

 to an ordinate, nor should it have any abi'upt curvatures. 



If we work by the trapezoidal rule, we may divide the base into any 

 number of intervals ; if by the parabolic (or Simpson's) rule we must 

 take an even number of intervals ; — and in either case we draw ordinates 

 through the points of division. Taking the simpler rule first — Avhich is 

 equivalent to assuming that the line joining the heads of two successive 

 ordinates is straight — the sum of the first and second ordinates is set off' 

 on the second ordinate. This represents twice the area of the curve 

 between those ordinates, w^hich doubled area is a strip equal to the length 

 so set off, and of the width of the interval between the ordinates. Twice 

 the area between the second and third ordinates is similarly represented- 



* An example of one arrangement for this purpose was given by the aiUhor in' 

 vol. vi. of the Trans. I.K.A. pp. 04-72, and also in Scott Russell's Naval Architecture, 

 pp. 135-138. It is a cumbrous process at best. 



B B2 



