V VII] Iiiiroduciion \ I \ 



The present table gives the first twenty tetrachoric functions, T O to r u> tal; 

 not to the argument T *=>k(l a A ), but to h itself, where h proceeds by intervals 

 of O'l and ranges from 00 to 4'0. It is, therefore, for many purposes more valuable 

 than the earlier table, but as the argument proceeds by greater intervals, it is 

 necessary to use interpolation formulae of higher order. 



The Biometric School defines the sth tetrachoric function r t (h) by the equation 



In place of the tetrachoric function, the Scandinavians have used the derivatives of 



' 



This practice suffers from certain disadvantages; it gives no simple nomen- 

 clature for the integral 



Jh 7T 



which we take as T O , or ^(1 a A ) of Sheppard's notation (see Part I, Tables I 

 III) ; and further it leads to much range of variation in the functions themselves, 

 so that the differences may be very considerable at one part and small at another 

 part of the table. 



The simple expedient of using (i) as the function causes all the tetrachorics to 

 be numerically less than unity. Lastly, (i) is the form which arises naturally 

 when we are seeking the volume (d) of a bivariate normal surface of total volume N, 

 of standard deviations cri, <TZ and of correlation r, cut off by dichotomic planes at 

 distances h<ri and k<r 2 perpendicular to the variate axes. For then 



^ = T Q (h)T (k)+rr 1 (h)T 1 (k)+r'T 2 (h)T 2 (k) + rT t (h)T t (k) + ...... (ii). 



This equation enables us to find d/N given r, h and k, or to find an equation 

 to determine r when d, h and k are given. The construction of triple entry tables 

 of d/N for r, h, k, now, however, very much simplifies the solution of either problem. 



If we wish to get the value of r t (h) correct to the seventh figure, then fourth 

 differences must be used in our interpolation formula; or, if we use central differ- 

 ences, we must take for 6 (= 1 <), 



T S (h + 0) = 6r a (h + 0-1) + <f>r, (h) 



120 



(ni). 



