350 



Mr. G. F. Becker on soi 



give in a footnote * below the details of the quadrature by (7) 

 of a portion of the ascending exponential. It will enable the 

 reader to perceive that no advantage is obtained by stating 

 the formulae in terms of the ordinates instead of the polygons, 

 even when the division of the area to be integrated is limited 

 to the minimum value of n. 



It is needless to say that the integrals (1) to (6) without 

 the corrective terms are rigorous for finite series with w + 1 

 constants whose highest terms contain x n . In any other 

 case two distinct means exist for reducing the error of the 

 result below a given tolerance, viz., a proper choice of the 

 number of derivatives to be eliminated and a subsequent 

 reduction of h so far as this may be needful. Even if 

 Euler's series ultimately becomes divergent for a given 

 function or class of functions, the earlier part of the series is 

 convergent and there is some term after the first at which 

 divergence begins ; in other words, the best result for a 

 given value of 7i is attainable by integrating a certain number 



# jr 



ind I e 2 



+2-6 



e x dx with n 



12, or h~0'2>, taking values of y from Smith- 



sonian Math. Tables, by (7). 



X. 



y- 



T. 



3 . 



4 . 



x = 



= -1-0 



?/o/2 





0-183 9397 







0-183 9397 



• r l 



-0-7 



y\ 



0496 5853 







x 2 



-0-4 



Vi 







. 1 



• r 3 



-o-i 



Vz 



0-904 8374 



0-901 8374 





*4 



+0-2 



y* 







1-221 403 



• T 5 



0-5 



y s 



1-648 721 







^8 



0-8 



lh 





2-225 541 





X l 



1-1 



y-j 



3-004 166 







x 'h 



1-4 | 



y% 







4-055 200 



X 9 



1-7 



y* 



5-473 947 



5473 947 





* l0 



2-0 | 



y\Q 









x n 



2-3 



?/n 



9-974 182 







*ia 



2-6 



yjt 





6731 869 



6-731 869 



2 .. 



Fac 



Pro 





21-502 439 



12-901 4634 

 = T 



15520 134 

 13-968 1206 



=o 3 



12192 412 



14-630 8944 

 = C 4 



or 



duct 





These values substituted in (7) give for the answer 13-095 858. The 

 true value of the last figure is 9. 



For n = 12, but not for any multiple of 12, (7) can be written 



*o 



ydx= I ] ^ ( yo+yi2 ) + 8( yi +^4-y 7 4- ? / u )4- j (y 3 +y 9 ) 



24 4 



4- y ye-jiy^+ys) 



[No arithmetical work is saved by adopting this method of statement.] 



