Asymmetrical Probability- Curve. 93 



III. The new proof of the formula for the asymmetric 

 probability-curve, which is offered here, is analogous to that 

 which has been given by Mr. Morgan Crofton for the sym- 

 metrical probability-curve*. The proof consists in deter- 

 mining y, the required error-function, as the solution of a 

 system of partial differential equations which must be satisfied 

 by such a function. Futy = Y(x, k, j) where ^andjhave the 

 same signification as before and k now= our c 2 -r-2, ( = Tod- 

 hunter's k 2 x 2). A first equation is obtained from the 

 condition that, if each of the constituent elements q 1} q 2 • . • 

 be multiplied by a constant y f , 



J =F( 7 « J y*, tV). 



Putting 7= (1 +o>), where co is indefinitely small, expanding 

 ami neglecting powers of co above the first, we have 



*+-£+»£ +*f-6. • • • (i) 



Two more equations are given by the conditions that, if 

 y = F(x, k,j) is the law of frequency for the sum of the n 

 elements Q/ q 1 + Q 2 ' q 2 + . . . , then the superposition of a new 

 element of the form Q'n+i q n +i for which the mean-square-of- 

 error (measured from its centre of gravity) is Ak, and the 

 mean-cube-of-error (similarly measured) is A;, must obey the 

 law of frequency 



y + Ay = FO, k + Ak, j + Af). 



Let v z= f[%) De the l aw °f frequency for the new element. 

 Then the law of frequency for the compound (of n + 1 

 elements) J is 



J> 



X0F(*-{) rff, 



where a and fr denote the extreme limits of the range of /(£ ) — 

 limits which are by hypothesis finite §. Expanding F(#— -f) 

 in terms of f , and neglecting powers of | above the third 

 (upon the hypothesis that the range of f is comparatively 



* In the article on Probability, ' Encyclopaedia Britannica,' 9th edit., 

 vol. xix. p. 781. 



t Of. Mr. Morgan Crofton, loc. cit. 



% According to the rule for compounding laws of error indicated by 

 Mr. Morgan Crofton in the article referred to. Compare the present 

 writer, Camb. Phil. Trans, vol. xiv. p. 141. 



§ Above, p. 90, and Phil. Mag. vol. xxxiv. (Nov. 1892). 



