TABLES OF POWERS OF NATURAL NUMBERS AND OF 

 THE SUMS OF POWEItS oK THE NATURAL NUMBERS 

 FROM 1—100. 



15v W. PALIN laDKRTON, Actuary. 



Ix many mathomatical iiive-stisj^ations it is of grcat advantage to havo a 

 table of tlie lower powers of the natural iiuinbors. This is proviiled in a haudy 

 form, as for as the second and tliird jjowers of tlie first 10,000 numbei-s are 

 coiicerned, in Barlow's Tahles (K. and F. N. Spon, London, Stereotype Edition). 

 But for many purposes we require to go beyond tiie cubes. Accordingly a table 

 has been prepared of the first seven powers of the natural numbers from 1 to 250, 

 and the first instalment of it 1-100 is given iu Table I. below. Only after the 

 work was completed did I discover that in the first edition of Barlow, such a table 

 was provided. My numbers were checked against his and found to agree. It was 

 still considered, however, desirable to publish Table I. below, because the first 

 edition of Barlow published in ISl-i is a most scarce work, and copies can rarely 

 be procured even at much enhanced price, after several years' hunting in second- 

 hand catalogues. 



With regard to tlic siuns of jjowers t;i1)le (Table 11.) Professur Pearson has 

 already published such a table from 1 to 20 with powers up to the 7th*. My table 

 ranges from 1 to 100 for the same powers. The final values of the sums of 

 powers table were checked against the sums of the powers of the first 100 natural 

 numbers as obtained by the usual finite differcnce method and this was also a 

 check on the whole of the powers table. Miss E. M. Elderton helped nie in sonie 

 of the work, which was done by the use of Crelle's Tables checked by an 

 arithmometer. As is well known the sums of the powers of natural inimhcrs can 

 be of grcat scrvice in loast s(juare investigationsf, and it is hoped that the preseut 

 tables will be found useful, not only to biometricians but to othei"s who ueed to 

 represent observations by parabolic curves. 



* Biometrika, Vol. ii. ]>. 10. 



+ Biometrika, Vol. ii. p. il et seq. 



