Past Experience on Future Expectation. 371 



Or, if we write € = (f>—q)/(n + 2), we can throw ft and @ 2 

 into a form which shows their approximation to the values 

 and 3 thus : 



*=-«-!? [ " + - H , " + 4 ' . (xiii.)'- 

 m(p + e)(q-e) m-1 y ' 



V m/ ^ »*. — 1 



/> + 3 

 6(m— 1) 



+ n + ?y 

 Hence if both m and n are large numbers : 



fl + 2-Y 



/3,= 



m(i' + e)('l-€) ljL m 



&=3 + 



3! 



&ts 



(XV.) 



(l> + e)(<l-e) m 



n 

 If //i be small relative to n, they become 



ft = . - ~"y -. and ft = 3H ? — — r-r- -. • (xvi.) 



^ ^(^ + 6)^-6; ^- w (p + 6 j(</-e) 



While if ?i be small relative to m, we have 



ft = f(?-P) 8 and £ =3+ ^ . ( xv ii.) 



These again reduce to still simpler forms if e = (^— p)/(n + 2) 

 be not comparable with either p or <J. i. e. if neither p or <[ 

 be very small. We then have for : 



(xvi.), & = (2~£)V("W), A = 3 + l/0w); 

 (xvii.), ft l -*4(?--#)V(»py) l A=3 + 6/(h|^). 



Both forms result. — for n or m large and the product of either 



2 D 2 



