214 Mr. W. Steadman Aldis. 



The relations (16) are adapted to logarithmic computation. The 

 logarithms of p v /x 2 , /* 3 , ... are given in Table VI. 



16. The verification of the values of K (aj), K 1 (a;) in Table II, can- 

 not be conducted on the method applied to those in Table I, because 

 the values of I (#), Ii(#) are wanting. 



A certain amount of check is given by the values of the four func- 

 tions I , Ij, K , K 15 calculated for the integral values of x, by the 

 former method, given in Table III. 



Two other checks, in addition to the useful one of performing all 

 additions and multiplications in two different ways, have been applied 

 throughout. 



The first depends on a very simple relation between the quantities 

 p r and /3' r . 



It is easily seen, from the general formula for the r+ 1th term in 

 (9), that 



Pr 3-5-21 ...... |(2r-l) 2 -4|. 



Pr '' l'3 2 -5 2 ....".. (2r-l) 2 



which, since (2r - I) 2 - 4 = (2r+l)(2r - 3), easily reduces to 



Thus ^ . ^ . H.- .................. (17) 



When the quantities p and p' have been calculated from the 

 logarithmic formulae, this result gives an easy method of verification. 

 It detects any mistake in the computation of the logarithms, or in the 

 derivation of the number from the logarithm. 



This formula leaves untouched the possibility of a mistake in the 

 value of P Q . To check this another process has been used. 



17. If f(x) be any continuous function of x, whose differential coeffi- 

 cients are also finite and continuous for the values of x considered, 

 Taylor's Theorem gives 



Let U Q be the value of f(x) corresponding to any particular value 

 of x, and let u v u 2t u 3 , ... denote the values of f(x + h), f(x Q 

 f(x Q + 3A), . . . Similarly, let u_ v u_ 2 , u_ B . . . denote f(x Q - h), f(x - 2h), 



Then !!LT.fz! = 



If h be so small that the terms of the series on the right hand not 

 written down may be neglected, and the three terms written down be 

 denoted by u, v, w, respectively, it follows that 



