380 Mr. L. Isserlis : Application of Solid Hypergeometrical 



The " fitting " of a double hypergeometrical series to a set 

 of statistics, say the frequency of a certain age of husband 

 and a certain age of wife, in a large number of marriages 

 will involve among the other things the determination of 

 r, r', n, and q in the corresponding chance problem. The 

 value of q and the ratios of r and r' to n will be numbers 

 characteristic of the particular type of frequency distribution. 



§ 3. Let 



[h-] k = h(h + l)(h + 2) . . . (A + &-1), 

 then 



("W [pn-r-r' + l] s+s , 

 If we put 



— r = at, — r' = a, —qn = j3, pn — r — r' + l=zy, 

 then 



^'* j ~ (n) r+r , " il«'![ 7 W ' 



Let F(a, a', /3, 7, #, ?/) denote the double hypergeometri- 

 cal series 



•7 s\s'\ [ 7 J 5+S , ' 'J ■ 

 So that 



22.0, 0= { -r$~ F(«, *', A % l, 1), 



\JI')r+r' 



or say =AFi(«, «', 0, 7). 



§ 4. We will begin by finding two partial differential 

 equations satisfied by F(a, a , /3, &, y) as a preliminary to 

 the calculation of the moments of the series. 



The coefficient of tl^MIll' in F( ^ a ' ? fr % Xy ^ 



is T W,[/3 + 5'] s .i' s „. , , 



' *1[7 + «V =*(«»£ + ' '? + «»«) 

 = X s - say. 



Similarly write Ys = F(a', /3-fs, 7 + 5, ?/). 



