on some Questions in the Theory of Chances. 151 



1 . 2 Wg 1 j'x"'dx. 



"^""^ -^ fx^-'{\-x)dx ^ 1.2 ^ -^ /x'«-^'(l-a;"-)rfa: ^*"^- 



/ 



This intearal beingr taken from x = o to x = i , 



"O"*' ^v.i..j, 



IS 



I Wa, W2-1,OT2— 2 l.m+1 



\m-\-n^ + 1 .m+«i +2...m+??i +n,+ l 



(Wi +W.3) (Mi+n^- 1)...(W| + 1) I Mj, Ms- 1, »J2— 2 l.m+1 



1 . 2 Wo 



m \ r W, + 1 .«2 .n,- 1 2 



+ ine'"--^/. — ; — -— — -^ — — - — m+\ .m + &c. 



_ («i + m) (wi + W2— 1) (wi + 1) 1 



~ 1.2 Wa m + 2 .m + 3...m + ni + nt + I 



{Ui.ns— 1 .112 — 2.. .m +ni.m + n, - I ...m + I e'" + n-^ + I ... 

 2 .?n + ni-l...m + 1 .m.m e"'~'/ + &c. 



rpu- • • 1 ^ d"^^'''.i/"^x"'(ex+fy)'" , , 



Ihis series is equal to dx"'d "■> — "^^-j when x and y 



are made equal to 1, and this is equal to 1 .2 .3 Wj. 1 .2 .3 Wax 



coefficient of ^"'Ar"", in the development of 



(1 + /i)"' (1 + k)"' (1 + e/t +/A-)"' 



( 1 + e/i +//t)"' = 1 + m (e A tA) + '"'"'"^ [eh +fkf 



1 . 2 



, m.m — I .m- 2 , , „,., ^ 



+ — r72T3 — ^^^* ^-^^^ + ^^• 



+ , „ „ A'-'-^/t", + &c. 



+ n2A''' A"'-' + w.n.A«.-' /£"»->+ "^ •"'•"' ~ ^ ^"■-»/c".-' + &c. 



1 • 1 • ^ 



