238 Prof. K. Pearson on certain Properties 



Now let a new series of functions xoy %\> Xii & c - De 

 formed, so that 



andletx =F(a, /3, y, a). 

 Then we have 



v s = c s x(xJXo) x= r 

 fi s can then be found from v s v 3 _ v v s _ 2 , &c. by the formulas 

 given on p. 77 of my memoir (Phil. Trans, vol. clxxxv.). 



Thus the determination of the successive moments of the 

 hypergeometrical series F t is thrown back on the discovery 

 of the %'s from the value F. 



3. To find the successive %'s. — The hypergeometrical series 

 is known to satisfy the differential equation 



x d / d¥\ . ,„ . x dF ^ . 



(see Forsyth, ' Differential Equations/ p. 185). 

 But dF 



Hence, substituting and rearranging, we have 



(l-^){% 2 +(m 1 -2)xi+(m 2 -m 1 + l)%o}+wXi-( n - , -^)Xo = 0. (6) 

 Put #=1, we have 



(%i)i=-^- (Xo)i (7) 



or rc + m 2 /ON 



This is the distance of the centroid- vertical of the hyper- 

 geometrical series F 1 from the vertical about which the 

 v-moments are taken. 



Multiply (6) by x and differentiate, we find 



(l-^){^3 + (m 1 -2)x 2 +(w2-^i + l)%o}-^{X2+( m i-2)x l 



+ (m 2 — m x + l)x<>} + n Xi - ( n + m 2>Xi = 0, 

 or 



(1-^){X3+ ( m l~l)%2+(^2-l)%l+(^2~Wl 1 + l)Xo} 



+ ( 7l - 1 )X3-( n + m i + w 2-2)x 1 -(m 2 -w 1 + l)%o = . (9) 



