148 Mr. Lubbock on the Calculation of Annuities, and 



multiplied by the coefficient of x" in the develoinnent of [x.+y)"'^", 

 the integrals being taken between the same limits as before. 



These integrations give for the probability required 



{m,+ \) {m, + ^)...{m, + n,) {tn, + m.,+7n^...+p - \)[m,+m,^m^...+p) 



(m.+OTa +Wp+P) (Wli+Wa +mp^j)+l) 



(Wig + WZ.i + OT^ +P + N-2) 



{nh + m^+m^ +p + n,+N- l)' 



(wi+1) (Wi + 2) (w,+iV) 



C bemg equal to — — ^ jy+i ' 



Adopting the same notation as before, this probability is equal to 



[nil + !]"■ [mi+ms... +m.p+p- ij 

 ^ [/«! + iiu + »«3 m^ + ri"' "" " 



_ C[»ii + l]"'[>«2+W3 + '«p+;j-l3"''"' 



which probability, as before, is the same as if the simple probability 

 of drawing a ball of the 7/" colour were wi, + i . 



If nh + ;«, + m,j + p-i = M, and if ni and A' are in the same 



ratio as wii and M, the chance that the niunber of balls of the 

 first colour in »/, + N trials is between the limits n, and h, ± z, by 

 the reductions given in the Thdorie Anal, des Prohahilith, 

 p 386, is 



c being the number of which the hyperbolic logarithm is unity, 

 and the integral being taken from z = z, to z - infinity. 



The question of determining the probability that the losses 

 and gains of an Insurance Company on any class of life are 

 contained within certain limits, is precisely similar to this. 



