L22 <>'. /•'. Beoker — New Meohanioal Quadratures. 



sives 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 (/' + 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 remain- 

 der 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 h into two or more parts.* 



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

 bers 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 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 argu- 

 ment. 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 (2). When experi- 

 mental data are to be dealt with observations can usually be so 

 arranged as to fit a duodecimal formula and when time is the 

 independent variable duodecimal 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 coefficients 

 of the higher derivatives as of the lower ones, and that a for- 

 mula similar to (6) could be found in wluch v' and v'" should 

 be retained but u xl and v xiiI cancelled. 



Odd values of n lend themselves less readily than even ones 

 to the elimination of derivatives from Euler's series because of 

 their limited divisibility. If n = 3 the quadrature may be 

 written 



/ 



X " , 9C, - C 3 o7 h'v'" 



1/dx = ! ? 3A. : 



8 240 



and the first term of this formula was given by Newton. f 



*The remainders in formulae (1) to (6) arise from the remainders in the 

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

 lished 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. 



f Opuscula, Methodus differential, prop, vi, Scholium. 



