No. 540] SHORTER ARTICLES AND DISCUSSION 757 



tions of x. In practise many cases arise where all the base ele- 

 ments of the observational curve are equal and the values of x 

 run in ordinal units from 1 to whatever number the observations 

 comprise. In such cases, taking the origin of x at 0, the sums in 

 (iii) which involve x and x 2 may be read off at once from Elder- 

 ton's 4 tables of the sums of the powers of the natural numbers. 

 If, now, similar tables are available from which one can obtain 

 the values of 8 {log x), 8(x log x) and £(log x) 2 for integral 

 values of x, there are left only the three sums in which y is in- 

 volved which must be directly calculated. 



So far as we are aware no tables have hitherto been published 

 giving the sums of these logarithmic functions of the natural 

 numbers. Consequently the present short table has been pre- 

 pared. The immediate incentive to its calculation was the fact 

 that in studies on growth and related topics in this laboratory 

 it has been rather frequently necessary to fit these simple loga- 

 rithmic curves. The table was calculated several years ago 

 purely as a labor saving factor in the work of the laboratory. It 

 has been used in manuscript here since that time. It seems de- 

 sirable to publish it in order that other workers may have the 

 benefit of the time and effort which it saves in curve-fitting work 

 of this sort. 



2. The values of S(\og x), S(x log x) and S(log x) 2 given in 

 the appended table were calculated twice independently, once 

 with 10-place values of the logarithms, and once with 7-plaee 

 figures. The 10-place logarithms were taken from Vega's 

 Thesaurus, 5 and the multiplications and summations were per- 

 formed on a large size Brunsviga arithmometer. As was to be 

 expected, the values of S(x log x) and S(\og x) 2 for the higher 

 numbers, when calculated from 7-place logarithms, were not ac- 

 curate beyond the fifth place. This 7-place table merely served 

 as a rough check on the accuracy of the 10-place work. The 

 tabled values given in this paper were all obtained by cutting off 

 the last 3 figures from the values in the 10-place table. The ac- 

 curacy of these last figures was previously tested by differences. 

 The table as given is believed to be accurate in the seventh place. 

 This is entirely sufficient because, as a matter of fact, in practical 

 curve fitting work one will not ordinarily use more than 4 or at 

 the most 5 places of figures in the logarithms. 



* Biometrika, II, 474-480. 



1 For the loan of a copy of these tables we are greatly indebted to Dr. H. 



