and the Elimination of Chance. 315 



attained the value ( ^ ) . We thus conclude that the proba- 

 bility-curve disengages itself from the binomial locus long- 

 before the extreme region of the locus is attained, and sweeps 

 on to infinity above the binomial locus. The approximation 

 is therefore safe when it ceases to be accurate. 



a ft 7. The conclusions which have been proved for sym- 

 metrical binomials may be extended to the whole class of 

 symmetrical finite facility-curves. Here we have not in 

 general the advantage of starting from an explicit* knowledge 

 of the probability that any particular degree of divergence 

 would occur — the general term of what may be called the 

 compound facility-curves. An explicit form may, however, 

 be attained for some of the simpler cases by the method of 

 Lagrange, which Mr. Todhuuter cites f. Consider the sim- 

 plest case of all (after the binomial), that in which the 

 elementary facility-locus consists of a rectangle of base 2a, 



and therefore of height x ■one value being as likely as 



Za 



another between the limits ±a. Lagrange gives the facility- 

 curve, which is formed by adding n values taken at random 

 from under this elementary facility-curve, or according to this 

 law of possibility %. The general expression is somewhat 



* Cf above. t P. 301 and following. 



% Lagrange's theorem may thus "be proved by way of an equation of 

 mixed differences. Let Mr. Todhunter's b=0, and accordingly his c = 



hist; = our 2a. Let his K = our g-. The primary facility-curve con- 

 sists of a rectangle whose base is 2a and height «-, and whose left base 



angle is at the origin. The compound facility-curve extending to the 

 right is such that its ordinate u sx represents the probability that, if s values 

 be taken at random from the primary facility-locus, their sum will lie 

 between x and x-\-dx; s and x corresponding to Mr. Todhunter's x and z 

 respectively. Observing the transition from u S x to u s +i, », we have 



I r x+2a 

 5- J* UsxdX = Us+\,x+2a' 



Put T) x U = Ux+2a', D s W = Ws+l. 



Differentiating both sides of our equation, we have 



[l-D7 1 ] s </)(.r)= (cf. Boole, 'Finite Differences,' 



chap. x. art. 2) 



(^y(0 s {^ ) ~ s ^~ ?a)+£ w ) ft*- 4 *)-**- } 



Observing the initial case when s= 2, we shall find that this expression 

 reduces to Lagrange's. 



\clx) \2a) 



