152 Mr. Lubbock on the Calculation of Annuities, and 



«..«.-! ^,, ^„,.. ^ n,.n..n^.-l ^„__ ,^„ _, ^ ^^ 

 ^1.2 1-2 



Coefficient of A'.r^ = i +ot (the + mf) 



m.m—\ /M, .ni- I » , „ „ „/• , "g-"^- 1 /-jN , o^^ 

 + 1.2 C 1.2 g +^"'"^^/+-TTT"'^y + ^^- 



The probability required is 



1 ■ 2 .3 Wi + OTa 



m+2 .m + 3 m + Tti + m,, + 1 



f , ., TW.m- 1 fn, .M,— 1 „ „ ^, W2.M2— 1 ^i 



Jl +m(>i.e + w,/)+ ^^ 1^2 c^ + 2M,M2e/+ ^^ /.} + &c. 



If there are p different colours, and if m trials have taken 

 place, and e,,p is the chance that a ball of the p"" colour was drawn 

 the 9'" trial, the probability of drawing rii balls of the first colour, 



H, of the second, n^ of the /»"" in Wi + n^ + n^ future trials, may 



be found in the same way. 



Let e,,, + e,,2 + fi,3 + &c e,,„ = 5,,ei, 



Cm, Ci.o + ei.3, e,.4 &c. = S^, e, 



(the sum of the products of e, two and two together,) 



Ci, 1 , 62, 2 + Ci, 2 , 62. 3 + &c. = Si e,, S, eo , * 



and so on ; then it may be shewn that this probability is equal to 



1 . 2 . 3...W1 + Wo + M3 + n„ , o v„ /, o v„ /, o ,„ 



— ' ; ^ (1 + Se,) • (1 + Scj)". (1 + .Se„n , 



1 + (Se,), 1 + (Scj), &c. being expanded by the binomial theorem, 

 and the indices of S written at the foot. 



* This is a method of notation which obtains, but it is not meant to imply that 



