liixii Tables for Statisticians and Biometricians [LI V 



The function H (r, v) is introduced because, as a rule, its logarithms have far 

 smaller differences and it is thus capable of more exact determination from a table 

 of double entry. Its physical relation to the curve may be expressed as follows ; 

 let the origin be transferred to the mean, then if y 1 be the ordinate at the mean, 



*-!*(b (xci)bis - 



where a is the standard-deviation of the curve 



a . ... 



= , (xcii). 



vr — 1 cos <f> 



The distance of the mean from the origin is given by 



/*/ = — a tan <f> (xciii). 



When r is fairly large : 



cos- 1 



/ ~ 5 T7T- 0'' tan 



1 / r e 3r Vir . 



jr(r,»)"V 2tt (cos0) r+1 " (XC1V) - 



TT 1 / V 1 -Aff2 / N 



Hence =a/ x -i—e ^ (xcv), 



H(r,v) Vr-1 V2tt 



, /l-4cos a d> 



where g=I ^/ _____£, 



and thus the evaluation if <£ be > 60° may be made by aid of Table II*. 



Illustration. In the curve fitted to the statures of St Louis School Girls, 

 aged 8 (p. lxxx), we have 



if- 2192, a =149917, 



r = 30-8023, v = 4-56967. 

 Find y„. 



We have tan <j ) = v /r= -148,3548. 



Hence <j> = 8° 26'-31315 = 8°-43855. 



Turning to the Tables, p. 136, we see the large differences of log.F(r, v) at 

 this value of <f>, and accordingly settle to work with log H(r, v). 

 We have for log H (r, v ), 



r = 30 r = 31 



$ = 8° -388,2032 -388,5583, 



<f> = 9° -388,2278 "388,5822, 



4> = 8°-4386, r = 30: 



log H (r, 1/) = -388,2032 + (-4386) [246] - £ (-4386) x (-5614) [28] 



= •388,2137. 



* For a fuller discussion of these integrals see Phil. Trans. Vol. 186, A, pp. 376—381, B. A. Trans. 

 Report, Liverpool, 1896, Preliminary Keport of Committee... , and the B.A. Trans. Report, Dover, 1899, 

 already cited. 



