Series to Frequency D istributions in Space. 383 



« 

 The equations (1), (2) lead to the following : 



(i-^ 2 g~ + (ww^J, +(7-<n,)| -«^g -<^=o ; 



and substituting for the differential coefficients their values 

 as above, we get the difference equations 



(l-^)teo-3%io + 2%oo) + (l-^)[%ii-%oi-%io + %oo] 



-H(7-m 1 + U')(% 10 -%oo)-«^(%oi--%oo)-W2^%oo = . (5) 

 (1-2/)(%o2-3xoi + 2xoo) -Kl-2/Xxn-Xio-Xoi + Xoo) 



+ (7- »? + lyXXoi + X00) -«'i/(%io-%uo) -m 2 '?/Xoo = . (6) 



Put a=y=l, 



■'■ (7-^i- 1 )(%io-%oo)i-<%oi-%oo)i— wi 2 (xoo)i = 0, 



(7-^i'- 1 )(%oi-Xuo)i-^'(%io-%oo)i~W(%oo)i = 0, 

 or 



(w— r) (%oi)i + »''(Xio)i = O l - ? ' + »'' + » ? 2')(%oo)i. 

 Hence 



/%io"\ _ J ) 2 — ?ir' — nr + (// — /■) rn 2 — rm 2 ' 



\%oo^=y=i //(// — y— /-'J 



or 



P'lo^ 1 * n( n-r-r') )=< 1 + ^ ■ (7) 

 Similarly 



p' 01 =o'(l + r' ? ) (8) 



These values agree with those found by Pearson (Phil. 

 Mag. 1899, p. 238, eqn. 7). This must be true in general, 

 i.e. the "marginal" moments p' 10 , p f 20 , ...£>'« must have 

 the same values as Pearson's v u v 2 , ... v t found for the single 

 iiypergeometrical series. 



For let 



fe s = A S q + A^i -f- A. s g . . • + A ss ' 



_ H.D8J. . . M.[«WW , 

 ' »IW. + *!*'i[ 7 ] s+ , 



_ «l[7]. t +, " + » ![*+•> " K- 'J 



[«].[/3.] n( Y +«-i)n( y - «'-/3-i) 



" s![ 7 ], n(7-/3-l)Ii(7-« , -l + *)' 



