1894.] The Asymmetrical Probability Curve. 271 



curves, especially at low pressures, must be considered rather as a 

 convenient way of showing which dots in the figure correspond to 

 any given pressure than as an attempt at interpolation. 



IX. " The Asymmetrical Probability Curve." By F. Y. EDGE- 

 WORTH, M.A., D.C.L. Communicated by Sir G. G. STOKES, 

 F.R.S. Received June 14, 1894. 

 (Abstract.) 



The asymmetrical probability carve is the second approximation 

 the symmetrical probability curve being the first approximation to 

 the law of frequency which governs the set of values assumed by a 

 function of numerous independently fluctuating small quantities. 

 The curve may be written 



1 =*T 2 x 2x 3 \-\ 



= e c * * 5 - ; 



v/^c L c 3 \c 3 c*/J 



where i/Aa; is the number of errors occurring between x and a;-t-Aar, 

 ^12 is the mean square of errors, and j is the mean cube of errors 

 errors measured from the centre of gravity. This form is obtained 

 by completing the analysis which Todhunter, after Poisson, has 

 indicated (' History of Probabilities,' Art. 1002) ; and independently 

 by obtaining a general form for the asymmetric probability curve, 

 and deducing therefrom the Poissonian formula in the case when the 

 asymmetry is slight the only case to which that formula is applic- 

 able. 



Among the peculiarities of the asymmetric probability curve are 

 the want of coincidence between the arithmetic mean and the position 

 of the greatest ordinate, and the descent of the curve at one ex- 

 tremity below the abscissa the ordinate appearing to denote negative 

 probability. 



An important case of the general carve is afforded by the Binomial, 

 for which each of the independent elements admits of only two values. 

 The approximate form of the Binomial, obtained directly by Laplace 

 (Todhunter, 'History,' Art. 993), is deducible from the general 

 theory. The general, or multinomial, probability carve can always be 

 represented by a binomial. 



The principle of the asymmetric probability curve affords an exten- 

 sion of the theory of correlation investigated by Messrs. Galton and 

 Hamilton Dickson ('Roy. Soc. Proc.,' 1886, p. 63). The symmetrical 

 probability surface 



