XXXI"-*, LI a ~ 6 ] lntrwin,-t;,,,, clxxv 



Thus wu ha\. 



#, = 885,5966, #,= M.S!', m::, A'.-l-iM'. />;, 1 :u:Ui490. 



We have now to substitute 7s' 5 in (xlii) with n = 5. Tli<Tc resultH 

 -891,0094 {-251,0094 A' 6 - '.V 



= 001,5219. 

 Hence pf + e = '331, 6592, and = -552,705. 



It will be seen that E^ is still not close enough to A* 6 , so that we cannot put 

 both = E' t and obtain the quadratic fbr E' . The value of A" from equation (xxxviii) 

 is 1 '297,268, which causes e to be negative, and we move in the wrong direction 

 from our first approximation. 



(6) The " most reasonable " value p of p. 



We have seen that in certain cases without knowing the actual value of p in 

 the population sampled, we know that in similar populations p is distributed alxmt 

 its mean p with a frequency <(/>), and that to find the p which will make the 

 probability of the observed r a maximum, we must solve the equation (see equations 

 (xxxiv) and (xxxv)) 



d_ r (i-fp**^) dz =Q ..... 



dp J o (cosh z pr) n ~ 



We will limit our consideration of this problem to the supposition that we have 

 previously observed that p varies with a standard deviation a p about the mean p in 

 a normal curve. Since when p = 1 the correlation will be perfect, we will assume 



, - 

 so that </> (p) = const, x e m ( l ~ P)* . 



Substituting in equation (xxxvi 6 ^) and differentiating, we have the following equa- 

 tion to find p : 



where E^ = (rpf + ...(xliv), 



V(l-(r/5)*)cos- 1 (-r 2 /5 a ) 



and successive E's are to be deduced from 



n(\-(rp?)E n+l = (2n-\)rp + n ^ (xlv). 



* 



Since p is the quantity to be found, it is clear that we can solve this system of 

 equations by approximation. There are two chief cases which may arise, according 

 as we have considerable or very slight knowledge of the frequency distribution of p. 



(1) We know that the distribution is very close to a mean p, with a small 

 variance, i.e. m is small. Our first approximation therefore is p = p. Let us suppose 



p = p + i/r. 



