Karl Pearson and Egon S. Pearson 
139 
Applying these tests we find : 
Sss' {X To T„') = 1 -000,000, Sss' {X Ti t/) = + -000,001 , 
Sss' T, T,' ) = + -000,001 , S,s' (% T, t/) = + -000,003, 
Sss- (% T, t/) - - -000,002, ,S',,. (^, T, t/) = + -000,001, 
and Sss- (^,s. T„ T,/) = + -000,002, 
results as close as we should expect, when we take into account the fact that our 
^t's were only to five figure accuracy, and our products to six. 
The meaning of Table III should be quite intelligible ; namely, for example : 
-084 = -^ = -079,464 + -001,947 r - -002,229 r- 
+ •001,538r3 - •014,745r^ + •000,897?-' - -014,884?-'^ + (xxi) 
is the equation which will give the correlation coefficient r as deduced from the 
(3, 2) cell. If r be given any other value the right hand of the above expression 
is equal to the contents of the (3, 2) cell for a normal correlation surface of corre- 
lation coefficient r having the observed marginal totals. 
Thus far the arithmetic is absolutely comparable with that needed for Ritchie- 
Scott's " polychoric r." We should have to solve the 49 equations, and then 
calculate — the stiffest part of the work — the probable eiTors of the 49 correlation 
coefficients which are the roots of these equations. Using these probable errors as 
our weighting data, we should find a mean coefficient. Our purpose is to replace 
the weighting and the solution of the 49 equations by the solution of a single 
equation. It will be noticed that both Ritchie-Scott's and our methods have an 
undesirable limitation, for we both assume the marginal totals to be those of the 
normal correlation surface. Actually in our case we ought to treat the marginal 
totals as vmknown, or select h^, Aj, li-i, ... ki, ko, k,., ... kq' as well as r to give as 
closely as possible the observed frequencies. Now the r's and consequently the 
J"s and ^t's and ^Ts all depend upon the h's and /c's and the equations obtained 
by making 
Sss' [ ) = minimum 
do not appear to lend themselves to any reasonably brief system of solutions. We 
were compelled therefore to introduce the admittedly limited form of solution, i.e. the 
determination of the best normal correlation surface subject to the restiiction of 
its having the same marginal totals as the observed frequency surface. We con- 
sider this a practically necessary but none the less grave restriction. 
We next proceeded to determine the value of riss'/iiss' and (nss'Aw)^ foi' certain 
selected values of r in order to build up equation (xvii) and solve it by inter- 
polation. The values chosen were : 0-45, 0-50 and 0-55. These cover the range 
within which we anticipate the solution of (xvii) for r will lie. We need also the 
value of the numerator in (xvii), i.e. 
vss- = Ti t/ -f 2 r ^, T, %■ T,' + 2r- % Ts %' +..., 
for the same three values of r. These results are given in Table IV. 
