Tables for the Solution of an Equation. 205 



These laws give, for values of n not less than unity, 



2 ~dx ~~ Vi + ln-i j 



> 08)- 

 2nl n 



~ In_l - In + 1 j 



They are known and will be quoted as the sequence laws. 

 4. It can be shown that K n (x) is expressible in two ways in terms of 

 a definite integral, namely, 



K n (#) = ( - l) n Tfn + i}^) e~ px (p 2 -!) 2 dp ... (7), 

 V ff;\ / Jj 



/y 



By putting j? = 1 + - in (7), expanding the binomial and integrating 

 x 



the separate terms, another form can be obtained for K w (^), namely, 



where the series within the bracket can be brought to a close at any 

 point by means of a remainder term, which, after a certain point in the 

 series, is always numerically less than the next term given by the 

 general law of the series. 



5. It is now possible to explain the processes by which the Tables I 

 and II at the end of this paper have been calculated. The series 

 actually employed are, for the smaller values of x, the ultimately con- 

 vergent series (6) ; and for larger values the series (9). 



In the calculation of the A functions, the natural logarithms of x 

 are required. These the writer has taken from "Wolfram's table at the 

 end of Vega's ' Thesaurus Logarithmorum,' having, in the numbers up 

 to 20 and for the prime numbers up to 59, verified them to 30 places 

 of decimals by calculation. 



The quantity E has been derived from Gauss.* 



Using Gauss's notation 



it follows that E = log 8 + r/( - J) = log 2 + 

 * ' Werke,' vol. 3, p. 155. 



R 2 



