Asymmetrical Probability-Curve. 97 



the expression 6{x,k,j). Then if we put F x for the first 

 approximation, 



9 =¥ t +e ; £=iY+0'; &<=•; 



where F/, 6' ... denote partial differentials with respect to x. 

 Now, if 6 is small with respect to F 1 ( t ^), then the functions 

 being continuous, 6' will be small with respect to ¥i(x). 

 And. we may likewise assume that Q" and Q'" are small in 

 comparison with F/' and F/" respectively. Therefore in 



the expression for -~ it is allowable to neglect Q' n '. But it 



is not equally allowable to neglect 0". For considering the 

 expansion of y + Ay, viz. 



„ l Al d 2 F l x .d 3 F 



F + -M-l-z — -A;^- 



2 d^ 6 ^ <£c 3 



(above, p. 94), we could not be sure that the neglected 

 quantity AkO" is not of the same order as the retained quan- 



tity A/ -jrj-j the terms of the expansion forming a descending 



series. Rejecting therefore only 6 fff , we have approximately 



d 3 y_d s F 1 _ 1 =?/a*_*\ 



/ 3# X 3 \ 



If t a?/ 



rfy 11 =*/.3* 



2k I 



whence by equation (3), 



a\__\ 



dj " 6 v/2tt1; 

 Integrating, we have 



% being a " constant " with regard toy; which by equation 

 (1) must be of the form 



Vk + ih)- 



Put .7=0 ; then the first term of the value for */ vanishing 

 while the curve becomes symmetrical, the second term, the 

 " constant " %, must be the expression for the ordinary pro- 

 bability-curve, viz. 



1 =? 



— = — =. e-tb . 



Phil. Mag. S. 5. Vol. 41. No. 249. Feb. 1896. H 



