ON QUADRATURES AND INTERPOLATION. 321 



Report on the present state of hnoivledge of the application of 

 Quadratures and Interpolation to Actual Data. By C. W. 

 Merrifield, F.R.S. 



Chapter I. Introduction. 



„ II. Interpolation by known properties of the particular function : use of 



Taylor's theorem. 

 „ III. General considerations relating to the application of finite differences 



to interpolation and quadrature. 

 „ IV. Theorems of finite difEerences. 



Sec. 1. Common formulas of direct in- 

 terpolation and quadrature 

 by ordinary differences, 



„ 2. Inverse interpolation. 



„ 3. Equidistant ordinates, not dif- 

 ferenced. 



,, 4. Multiple integrals ; ordinates 

 not differenced. 



,, 5. Quadratiure by differential co- 

 efficients. 



Sec. 6. Interpolation of direction : 

 maxima and minima. 



„ 7. Symmetrical differences. 



„ 8. Definite, or tabular interpola- 

 tion. 



„ 9. Interpolation of double entry 

 tables, or functions of two 

 or more variables. 



Chapter V. Interpolation and quadrature with ordinates not equidistant. 



Sec. 1. Newton's method. 

 „ 2. Lagrange's method. 



Sec. 3. Gauss's method. 

 „ 4. Other methods and suppositions . 



Chapter VI. Interpolation and quadrature for uncertain values. 

 „ VII. Periodicity. 



„ VIII. Systematic computation of quadratures and interpolations, 

 „ IX. Graphical methods. 

 „ X. Mechanical quadratures. i 



I. — Introduction. 



The questions, both of interpolation and quadrature, will be considered, 

 for the purposes of this report, chiefly with reference to the two following 

 cases : — 



(a) Where a definite number of observations is given, and no inter- 

 mediate observations are procurable. This is the case with most 

 records of isolated or discontinuous observations, and witli time 

 observations. 



(b) Where a curve is mechanically or graphically given, either actually 

 or implicitly, so that ordinates can be taken at pleasure, while 

 the analytical expression of the curve is either unknown or not 

 available. This is the case with the graphical record of con- 

 tinuous observations, and with the calculation of areas and mo- 

 ments in engineers' work, and in naval architecture. 



When any varying quantity, or function, is tabulated, the table gives 

 the value of the function corresponding to certain given values of the 

 subject of the function. The values of the subjects are termed the 

 argumenis of the table : the corresponding values of the function are 

 termed the entries. Interpolation is the problem of finding the value of 

 the entry corresponding to an argument not actually given in the 

 table, but usually intermediate to the extreme arguments. When the 

 form of the function is absolutely unknown, except from the definite values 

 tabulated, interpolation is essentially an indeterminate problem. 

 1880. Y 



