1G 
PROFESSOR K. PEARSON ON MATHEMATICAL CONTRIBUTIONS 
either, but not both, b and c vanish, nor does it, unless we multiply it by tt/ 2 and 
take its sine, equal the coefficient of correlation when a = cl and b = c. 
Again, we might deduce a fairly simple approximation to the coefficient of correla¬ 
tion from the Equation (xxiv.) for 0, using only its first few terms. Thus we find 
where 
„ _ ad - lc _ 
‘ 111 1 - h(Xi + A/)} + (fid - be) 
(liv.), 
as an expression which vanishes with the transfer, and will be fairly close to the 
coefficient of correlation. It is not, however, exactly unity when either b or c is 
zero. But without entering into a discussion of such expressions, we can write 
several down which fully satisfy the three conditions :— 
(i.) Vanishing with the transfer. 
(ii.) Being equal to unity if b or c = 0. 
(iii.) Being equal to the correlation for median divisions. 
Such are, for example :— 
Qa = 
Qi = 
tv sVad — y/bc 
Sin--=--y=. 
2 y/ad — be 
sin fl+ 2 he' y ’ ad > bc ■ ■ 
(ad — bc) (b + c) 
(lvi.), 
where 
. 7T 
sm — 
I 
\/l + K 2 
_4 abccl N 2 _ 
(ad — bc)~ (a + d) (b + c) 
(IviL), 
Only By actual examination of the numerical results has it seemed possible to pick 
out the most efficient of these coefficients. Q x was found of little service. The 
following table gives the values of Q. : , Q 3 , Q i; and Q 5 in the case of fifteen series 
selected to cover a fairly wide range of values :— 
