402 Application of Solid Hyper geometrical Series, 



§ 13. Note to Phil. Mag., Feb. 1899, pp. 236-246. 



In fitting a hypergeometrical series to a frequency polygon 

 iby the method described in Professor Pearson's paper, certain 

 ambiguities of sign may be removed by the following con- 

 siderations. The tenns of the series (1) are the chances of 

 selecting 1, 2, 3, &c. black balls out of a bag containing pn 

 white and qn black balls in a draw of r balls. By equation (2) 

 m 1 =—r — qn must be negative, also in the real case 

 .r<pn or qn .*. —mi<n and a is .*. numerically less than /3. 

 Again, by (22) e=n 2 -\-nm 1 -\-m 2 .'. in general e>m 2 . By 

 •(23) V / A = V / ^ r T(2e 1 -n 2 )/(n~2) •£, but \/%=* ^1°* 



n 2 

 where a is the positive square root of //, 2 , .*. z± "^-^ according 



Hence (34) should be written 



*=i(W^fe?> 



the + sign occurring when /-t 3 >*0 and the — sign when 



/* 3 <0. 



(35) becomes: — m 2 , eare roots of ^ 2 —z 1 ^-{-z 2 = i e being the 

 greater root, and (37) becomes a, j3 are roots of f 2 — m L %+ m 2 = 0, 

 where a is numerically less than /3. 



As an illustration we may apply the formulae to the series 



1 + 24+ 90 + 80 + 15, 

 which is really F(— 4, —6, 1, 1), i. e. we snould obtain 

 *=-4, /3=-6, y=l,^ = 0-4, 2 = 0-6, r=4, n=\0, c = l, d = 2'4. 



We find 

 ^ = 2-4, yu 2 = -64, ^ 3 =--032^ 4 =l-1254f. /* 5 =--21961f 

 ^ = •00390625, /3 2 =2-747768, & = 10-72321. 



whence (32) gives 



_ 4&-10A + 6& + 2 

 n ~ A-4A + 3A + 2 -" JU - 

 From (33) 



, * 3 ("-l) _ 576 



^" 4(72-1) +2/3 2 (n~2)-/3 3 (n-4)-°^' 

 By (34) 



V n — 1 



the minus sign corresponding to the negative value of y^ 3 . 

 Thus ^= 48. 



