on some Questions in tlw Theory of Chances. 146 



multiplied by the coefficient of 



* ..... 



Since x, + Xo . . . . + oc^ = l, if x,, x.^, &c. be all .supposed to 

 vary from o to i, and all these values to be equally possible 

 a priori, the numerator will be found by integrating the ex- 

 pression 



.r, '"■ + "■ X Xo "'=+". (1 -x^-x^^-x^ -Xp_C)"'v^"pdx^ x dx^ dXp_, 



first from Xp_^ = o to Xp_^ = i— xi-j^2 —^p-i, 



then from Xp_2 = o to Xp_.2 = l — a?i -Xp_3, 



and so on. The denominator vt'ill be found in the same way. 



If the coefficient of Xi'"' x xj""" .... Xp'% in the development of 



(x^+x,.... +x^)».+«. •••• +"^, 



be called C, these integrations give for the probability required 



^,^ (wi.+ l) (y«i + 2)(mi+3)....(mi + Wi) (m^+l) im, + 2) ....{m^ + n^) . ... 

 {nii + mi + m^ +mp+p) {mi + mi + nis + nip + p + l) 



(mp+ 1) (?Wp + 2) (nip + np) 



(mt + m.^ +j9+n, + n2 4-^3- 1)' 



or if the product 



{nip +l)(mp + 2).... (nip + n^) 



be denoted by [nip + i]"p, which is the notation used by Lacroix 

 Traite du Calcul Differentiel, Vol. III. p. 121 ; the probability 

 required is 



C -. [^' + 0"' K + 1]"' ■ • • • [mp + l]"r 

 Vol. III. Part I, T 



