UQ Mr. G. F. Beck 



ecker on some 



differences appearing in the remaining portion of the series 



buTo"; eon ? SS f \° f fllnCti ° ,1S ^ ™* U be ^vantajeou " 

 but on account of the semi-convergence of Euler's series the 

 desirable limit will in many instances be lower 



1 have carried out the process only as far as n=12, whieli 



eWentl it e ' ,m r ti0 " of al J ^ derivatives below the 

 eleventh. The equations themselves show the appropriate 

 factors while the coefficient of the one derivative tern 

 retained is the sum of the coefficients in the several En er 

 series each multiplied by one of these factors. The following 

 six formulas are thus obtained :— ™ g 



n 



(1) fa = *l±Gi_2k 



+ . 



360 "• * > 2) 



9) _3»T + 12C,4-Gk i . AV 



~ls~ -- ih Im) + ---- (»>*) 



, 3 648 T + 81 C 2 + 112 3 -C 6 ,. , /,V« 



KJ 8«r _ "" oA 5:6do + ---- (" >6 ) 



(4 _ ^04ST + 704C 2 4- 84 C 4 -C 8 B1 2AV 



(.5) _ 35,000 T+ 14,375 C 2 +528 C 5 -7 C , ln ,10AV* ^ 



~«^b ^- 10/i 12l]W ( ' l>10 ) 



(6) = M92,992T- 174,960 C. + 585,728 C 3 



1,801,800 

 _ 104,247 C. 4 + 2,288 C 6 -C 12 , „ 691 A"^ 



TsOl^Otr -- Uh lWjM +••••(» >12) 

 The derivatives in the last or corrective terms of these 



Z'^Tl? TJ i 6 ex P ressed in ter ™ of finite differences 

 should the latter be more convenient. The transformation is 

 well known, but its most essential features may be noted 

 here to save a reference. The *th derivative of a function, 



/(>), multiplied by the Mb power of the constant increment 

 ot x here denoted by h, is expressible in terms of the *th 

 Unite difference and differences higher than the kih For 

 the purpose m hand Newtonian differences should be em- 

 ployed because they are applicable at the beginning and at 

 the end ot a series of values. When the derivatives and 

 differences are so large that higher derivatives and differ- 

 ences also require consideration the transformation is some- 

 what complex, but if the Hh difference is technically "small" 



