Ixxxii Tables for Statisticians and Biometricians [LIV 



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^ be the ordinate at the mean, 



N 1 . sbis 



y^=^H{rrv) ^^"^ ■ 



where a is the standard-deviation of the curve 



a , ... 



= 7^ (^"i)- 



V ?• — 1 cos <p 



The distance of the mean from the origin is given by 



fJn.' = - tan 4> (xciii). 



When r is fairly large : 



cos- d) 1 



1 / — -^:;^-To;.-'f"'^^'^'f' 



T(^)~y 2^ (cos </))'•+' ^''°'''''' 



Hence ^ = ./ -L- x 7=e"*''' (xcv), 



H{r,v) V r-1 V27r 



, /I — 4cos-<i 

 where 5 = ^/ ^^ , 



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. Ixxx), we have 



N= 2192, a = 14-9917, 



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

 Find y„. 



We have _ tan ^ = vjr = -148,8548. 



Hence ^ = 8° 26'-31315 = 8°-43855. 



Turning to the Tables, p. 136, we see the large differences of logi^(?-, v) at 

 this value of <p, 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, 



= 9° -388,2278 -388,5822, 



= 8°-4386, r = 30: 



log H (r, v) = -388,2032 + (-4386) [246] - ^ (4386) x (-5614) [28] 



= •388,2137. 



* For a fuller discussion of theee integrals see Phil. Tnnis. Vol. 186, A, pp. 376 — 381, B. A. Traits. 

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

 already cited. 



