184 <>. F. Becker H Mechanical Quadrafa 



a given tolerance, viz.. a proper choice of the number of deriv- 

 atives to be eliminated and a subsequent reduction of 1< bo far 



a* tin's may be needful. Even if Euler's Beriee ultimately 

 mes divergent for a given function or class of functions, 

 the earlier part of the seri - sonvergent and there i> some 

 term after the first at which divergence begins: in other words, 

 the best results for a given value of It is attainable by integrat- 

 ing a certain number of terms of Euler's series. This amounts 

 to a choice between formula? (1) to (6) or others similarly 

 derived. Thereafter the only question is how small a value of 

 h must be adopted to give the required accuracy. Any quad- 

 rature by series assumes a fictitious curve more or less nearly 

 coinciding with a real one. For the quadratures under discus- 

 sion the number of derivatives eliminated determines the order 

 of the contact of the two curves at the extremities of the arc to be 

 integrated and also a minimum number of common points on 

 the two curves. By division of h the number of common points 

 is increased in simple proportion to the number by which /< 

 mav be divided, but the order of contact at the extremities is 



r ind / e*6 



Find / e*d x with n = 12, or 7i = 03. taking values of y from Smith- 



sonian Math, Tables, by 





X 



y 



i 





C 





C 



4 



x i = 



-1-0 



- 







0-183 



9397 



0-183 



9397 





-0-7 



!/i 



0496 



5853 











• r » 



-0-4 



y* 















*» 



-01 



ifi 



904 



S3 74 



0904 



8374 







*4 



- 2 



y* 











1-221 



403 



** 



0-5 



Vi 



1-648 



721 











*, 



0-8 



y« 







2-225 



541 







. 



1-1 



y- 



3 004 



166 











■ 



1-4 



ye 











4055 



200 



X, 



1-7 



y» 



5-473 



947 



5 473 



947 







x lt 



2-0 



Via 















*11 



2-3 



Vii 



9974 



182 











X,, 



26 



Vi*& 







6731 



869 



6 731 



869 



V 



21-502 



439 

 27i 



15-520 



134 

 37i 



12 192 



412 



Factor 





47; 



Product 





12-901 



4634 



13968 



12o6 



14630 



8944 









= T 





= C, 





= c. 





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 



/'ydx = A j -^ ( j/o -f y„) + 8 (y, + y, + y-. + y tl ) + ^ (y, + y>) 

 5 ' , 



x» 



24 4 / , vl 



- . - =-(»« + »e) f ■ 



Xo arithmetical work is saved by adopting this method of statement. 



