W. F. SiiEPPAiiD 176 



numbers of individuals whose measurements are respectively loss and greatcr than 

 any particular value A' are i\^i and N.,, where 



and the ordinatc «f the figiire of frc(]uency corrosponding to the incasurement 

 X is 



Nz/a, 



where i(l +a) and z are the vahics of the area and of the ordinale given by 

 Tables I. and 11. as corresponding to 



« = (X — 7)1) la. 



Method of Using. 



2. Interpolation. If a quantity m be tabulated in terms of x, and (/„ bc the 

 vahie of u for « = «0, then its vahie for x = Xo + 6h, where h is the common 

 differeuee of x, and 6 is less than 1, is usually foiind from w,, and its advancing 

 differences by the formula 



«=.„+.A.„-^>A.. + ^-l^-)i^)AX- (3). 



An alternative formula, in which central differences are used, is that recently 

 given* by Professor Everett. When we do not require to go beyond third 

 differences, this formula is most conveniently written 



u = «„ + ÖA«o - -^^^ — ^ S'"i - ^^ g/ ^ S"«o (4), 



where S'wi and B-tio are the second central differences of u^ and Wj (i.e. B-Ui = A'u^, 

 8^0 = A-(/_i), aud (/> = 1 - Ö. 



In practice, when second and higher differences have to be considered, and a 

 mechanical method of porforming multiplications is available (e.g. by means of a 

 Brunsviga), it is more convenient to use differential coefficients. Let the means 

 of the pairs of odd differences above and below the line of ii„ be denoted by 

 /j,Su„, fiS'iio, ... (i.e. /^8(^, = -^ (Ai(„ + A<<_i), /u.S'it„ = |(A''i6_i + A'fLj), ...), and let 

 Dt(o, B-Ufi, IPuii, ... be the values for x = x^ of the successive differential coefficients 

 of It with regard to x; theuf, 8^ having the same raeaning as in (4), 



IiDua = /i8»„ - ^ fiBhi^ + ^Vm^^^'o - •■•\ 



hWui) = S-u„ — f\B*u„ + ... 



hWUa = flBhlo-^/jl,B''Uf, + ... )■ (5), 



h*D^Uo = B%- ... 



etc. 



and, for x = Xo + dh, 



u= «0 + [hDu„ + \e [h'D-u, + {6 (hWiio + ...)]] (6). 



* Journal of the Institute of Aetuaries, Vol. xxxv. p. 452. 

 t Proe. of the Land. Math. Soc. Vol. xxxi. p. 465. 



