W. F. SllEl'PARD 



177 



A first approxiination gives 



= 14217^36827 =-38605; 



and, with this value, the correctcd divisor becomes 



36855-2, 



which give.s f'or a second approxiination 



= -38070, 

 and therefore 



« = •39614.2.5. 



The conoct valuc, as givcii hy Table III., is 



,6 = -3961424. 



The degree of accuracy with which x can be obtained by this method depends on 

 thc relation of maguitude between the differences of x and of ii. In the above 

 example, to a difference of 'Ol in x there conesponds a difference of veiy little 

 more than one-third of Ol in ^(1+a); and therefore, if *• is calculated from 

 Table I. to seven places of decinials, it will only be accurate within about 2 in the 

 lastfigure. The possible inaccuracy of *• increases as ^{1 + a) increases. But this 

 is not importaut, as the " probable error " of oc, for any given minibcr of observations, 

 also increases. 



4. Smoothiiig. In arraiiging a table, with difierences, for the calciilation of any ciuaiitity v, 

 it is usual to enter in the iliti'crence-cohimns the actual (or " tabiilar " ) diticrences of thc vahies 

 of u as tabidated. In the prosent tablcs I have adopted a different method, and ha\-e given the 

 diflferences as near as possible to the differences of the true values of u. The object of thi.s is to 

 enable greater accuracy to be obtained when required. If we only want u to tive or six place.s 

 of decinials, it is imniaterial whether we use the tabular or the corrected dift'orences. But, if we 

 wi.sh to have it as accurate as possible, we can alter the tabulated values by inspection. 



Looking, for instance, at the commencenient of Table L, it is clear that the tabulated values 

 of i(l+a) are too great for .);='01 and .r=-03, while they are too small for .r=-02 and .r = -04. 

 Taking M=i (1 -\-a) x 10", so as to omit the decimal point, the table may be written 



By tabulating v, by differences of -05 or -10 in .r, it will be found* that the third difference in 

 Table L, for these values of .r, is almost exactly 4. We see therefore that ö + c^ and x + ^ 

 are both greater than i, and (^+x is less than \; while ö, \-6-<i>, 4> + x^ ^^'^ ^~X~^ ^rs 

 all very nearly equal. The values ö=-4, <^ = -2, j( = % V'=^i satisfy the.se conditions ; and, as a 

 matter of fact, they give for i (1 + a) values which are correct within 1 x lO^**. 



* For the relations between diflferences of w for large and small differences in .r, see Proc. I.nnd. 

 Muth. Soc. Vol. XXXI. pp. 46S— 471. 



Biometriliii ii '.^3 



