386 Mr. L. Isserlis : Application of Solid Hyper geometrical 



To find p 2 \ and p 12 we again apply (13) and (14), and 

 obtain 



[^(l^P + Q]! =[-0P + 0<ftQ]„ . . (26) 

 [^(l- < z/R + S)] 1 =[-^R + ^S] 1? . . (27) 

 leading to the equations 



— %2i — %i2 — %n (™i — 3) + % 2 (r + 1) - % i (™2 — ™i + r + 2) 

 + ( n _^j %21 + r%12 + %n (_ r _ m2 + r '_ n )-0. . . (28) 



-%i2-%2i~%ii(V-3)+x 20 (r' + l)-% 10 ('m 2 '-??ii' + r' + 2) 

 + (^-^)%i2 + ^%2i + Xn(-r'— m 2 ' + r~n)=0. . . (29) 



These equations are equivalent to the following for the 

 " raw " moments 



*=y = l 



P21 



P\2 



P'll 



P02 



(n-r'-l) + ^3|(r-l) + ^( r '-» + 3 + ^-r^)+ ^(r + 1) 



P'01 / 1 



^_21 

 € 2 C 



?i(r'^ + ^ + 2/ + 2) = 0, . (30) 

 \(r'-l)+^ 2 (n-r-l) + ^(r-n+3 + qn-r'qn)+^f(r' + l) 



(£C Cw C 



-^{rqn + qn + 2r + 2} = 0. . . (31) 



Now if d=(l-\-rq)c^ d' = (1 + r' q)c' denote the position of 

 the mean, the raw moments can be expressed in terms of the 

 moments about the mean by means of the equations 



p'21 =P2i + 2dp n + d'p 20 + d-d 

 p'12 =Pi2 + 2d'p n + dp 02 + dd 12 

 p\i=Pn + dd' 



P'02=P02 + d' 2 

 p'20=P20 + d 2 



Making these substitutions in (30), (31) we get after 

 rearranging 



(32) 



J 



P21 



c 



|(„-/-i) + gl(,-i) + ^^-/-i) + ^(,-i )+ r+i) 



d 2 d' , 1N , dd'* ,. dd' , , „ N d'\ „ 



+ oV (n ~ r 1)+ "^ (r ~ 1)+ ^ r(r '-n + Z + qn-rqn)+ V2 (r+l) 



d' 



— — (rqn+qn + 2r + 2) = 0, 



and a similar equation 'with suffixes and accents inter-changed. 



(33) 



