398 On the Tabulation of certain Frequency- Distributions. 



constants. Consider, for instance, the probable error in M m , 

 when both the bounding ordinates are infinite. Let N be the 

 total number of observations. Then, if we knew the values 

 of X exactly, the probable error in M m = 2X" l /N would be* 



67449.... V(M 2m -M^)/N (9) 



s error in the expression on the right-hand side 



67449.... n m /s/N, (10) 



where 

 2m 2 r x a»-a X^- 2 X 2m " 2 



X 2 " 1- 2 



~\Tm—\~STm — \ Vin-lYm-l "Vm— lVm- 1 



+ 8(j,-8) ' f» +2(y-4) \f* + .,. + a^3=L 



x»- i xr- x x-^x™- 1 x^x™-, 1 



+ 3(j>-4) 3 7 4 + 3(j>-5) 3 yr/ 5 +-. + 3 ' p ~ l 



+ (p-2).L. v-^p-^ K (n) 



Lip—lLip-X J 



We may regard (11) as the approximate expression, by a 

 quadrature-formula, of a more accurate value of £l m . This 

 value is to be found by making p infinite, which gives 



ni = 2m*\ A^-| J (1-A)- Z - dkjdk 



= %\ A.mX—'l f (l-A).mX— 'rfxjrfX; (12) 



and it is not difficult to show that this last expression is 

 equal to 



Mo — M 2 

 which agrees with (9). 



* For the method of finding the probable errors considered in this 

 section, see Phil. Trans, vol. 192 (A) 1898, pp. 124 seqq. 



