310 PROFESSOR K. PEARSON AND MR, L. N. G. FILON 



[NOTE. Added May 25, 1898. One point ought to have been more fully dealt with 

 in the above memoir, namely, the probable error of the criterion K = 6 -f 3j8, 2fi.,, 

 upon which the selection of the type of the frequency depends. Clearly, if the 

 probable error of this criterion is as large as the criterion itself, there can be no 

 stability of type, or the frequency may change over from one type to another. 



On page 289 we have found the standard deviation of the criterion in terms of 

 known quantities for the curve 



It is in fact given by the umbral equation 



X.A/K = tXjfe, + i.&.jG.. ....... (clxv.) 



where i } and I, are functions of m l and m., given in (cxx.) and x,X/. = ^>,, ' 8 known 

 from (cvi.). 



The standard deviation of the criterion for the curve of type 



y = y e--*l{\ +(aj/o) s } M 

 may be found by taking differentials of 



K = 6 + 3/3, - 2& = - ^~ j 4 ^n^~~ + 1 J , 



a value readily obtainable from 'Phil. Trans.,' A, vol. 186, p. 377. We thus find 

 the umbral equation 



^ J , . , , i* - 3 ; + 1 12 1 96 sin cos <f> (r - 1) ^ 



XA - {96 sin- ^ jj + (7 _^J 2 , Xr -- ( , _ 3) (r _ 2)3 - S, X , (clxv,.) 



where 2,, S^, and x/X* ^ are gi ven by (clii.) and (cliv.). 

 Applying these results to the numerical examples, we find : 



(a.) For the glands of swine 



^ X. = - -070,5459 2 WiXMi - -038,4629 2^ XlBi , 



K 



whence the probable error of K = '67449 2* = '1012 ; or, 



K -5559 '1012. 

 (b.) For the stature of children 



S.x = -01 5,6206 S 7 . Xl . - -018,0067 2,x 

 whence the probable error of K = '67449 2 K = '1919 ; or, 



K = - -4330 '1919. 



