322 BEPOET — 1880. 



When the form of tlie function is known analytically, and can be used 

 for the purpose of deteimining the intermediate value or values required, 

 the problem becomes determinate, but the work is then rather that of 

 com]pufation than of interpolation. This is equally true whether the 

 computation be direct, and independent of the table, or whether the tabu- 

 lated values be used to facilitate the computation of those not tabulated. 

 The latter case, under a slight change of aspect, is usually included in 

 the term interpolation — namely, interpolation by means of the known 

 properties of the particular function tabulated. This is not included 

 in the general problem of interpolation, which is the object of this 

 rejiort. 



The method of quadratures is usually understood to mean the inte- 

 gration of a function by the use of certain definite values of it. The 

 geometrical expression of this is the quadrature of an area by means of 

 its ordinates. There are two principal and distinct cases of this — one 

 where the function or curve is only known for certain definite values or 

 ordinates, and not intermediately, and the other where the function or 

 curve, although not analytically given, so that the integral calculus can 

 be applied to it directly, is or may be known at any selected point or 

 ordinate whatever. The first case has indeterminateness of the same 

 order as the corresponding problem of interpolation : the second presents 

 itself in the case of curves actually drawn or otherwise continuously 

 indicated, and practically also where the function, although given in 

 analytical form, is not the differential coefficient of a function which can 

 be directly computed ; — and this second case has, in itself, nothing inde- 

 terminate. 



Interpolation has also to be considered with reference to differential 

 coefficients as well as to the function, and also with reference to maxima 

 or minima either of the tabulated function or the argument. 



Quadrature also has to be applied to moments as well as to simple 

 integrals. Multiple integrals have also to be considered, but, like simple 

 integrals, always between constant limits. The principal types of these 

 are, in addition to the simple integral, 



/: 



y dx 



the moments / ^ xy dx 

 xhj dx 

 y^ dx 



f' 



/: 



"* y^ dx 



/: 



and multiple integrals of the two types 



flflft «^^^* 



and 



