348 Mr. G. F. Becker on some 



(r+1) 



any case whatever let X be the maximum value of f(x) 

 between the limits, then the total area represented by the 

 definite integral, v r , must be less than \(«r» — a' ), and this 

 substituted for v r in the corrective term gives a quantity 

 greater than the remainder of the Eulerian series. It is 

 usually practicable so to select or subdivide the limits 

 of integration that the (r + .l)st derivative neither changes 

 sign nor passes through a maximum, and then the corrective 

 term of the equations approximately defines the error of the 

 quadrature. 



It is possible still further to reduce the limits of the 

 remainder provided that certain assumptions are made with 

 respect to succeeding differential coefficients, but this proviso 

 implies an inquiry which in most instances would be 

 laborious, more so than the division of li into two or more 

 parts *. 



It will be observed that even in (6) the coefficients are 

 numbers of manageable magnitude not exceeding 7 places. 

 If the attempt were made to eliminate a larger number of 

 derivative terms it is not improbable that the formulae could 

 be dealt with only by 10-place logarithms or computing 

 machines. 



The most accurate of the equations given above involves 

 the division of x n — x Q into some multiple of 12 parts. 



Such a division may be inconvenient, for example in 

 dealing with a function which is already tabulated to a 

 decimal argument. This difficulty, however, may be avoided 

 with little trouble ; for, if 100 values of y are available, the 

 quadrature from the first to the 96th may be effected by 

 equation (6) and that of the remaining 4 items by equation 

 (l'). When experimental data are to be dealt with observa- 

 tions can usually be so arranged as to fit a duodecimal 

 formula, and when time is the independent variable duo- 

 decimal division is of course most convenient. 



In dealing with some functions one or two derivatives are 

 readily calculated, or are perhaps already tabulated, while 

 the higher derivatives are troublesome. It is worth while to 

 observe, therefore, that it is as easy to eliminate the coeffi- 

 cients of the higher derivatives as of the lower ones, and that 

 a formula similar to (6) could be found in which v' and v'" 

 should be retained but v* 1 and t> xiii cancelled. 



* The remainders in formulas (1) to (G) arise from the remainders in 

 the system of Taylor's series on which Euler's equation is based, and the 

 published discussions of the remainder in Taylor's series would make a 

 stout volume. No elaborate consideration of this subject seems needful 

 for the purpose of this paper. 



