20 
PROFESSOR K. PEARSON ON MATHEMATICAL CONTRIBUTIONS 
Now n x and n. 2 ' will be as the angles between the strained positions of the planes 
bounding n 1 and n 2 . Ox does not change its direction. Oy is turned through an 
angle 7t/2 — D clockwise, and y becomes y", say. Hence 
or 
< : %' :: I - x" + \ ~ D : 1 + X ~ \ + D 
(%' - <)/(%' + O = | ix" + H) - 1. 
Let us write E x = 2 (n. 2 — n 2 ) and term it the excess for the y-character for the 
line AB. Then we easily find : 
/E 1 vr tt\ / // . -p. N cot x" + cot D . ... 
tan (n 2 + d = tan + p ) = eot X " cot D - 1 ■ ■ ■ ■ < lvm ')- 
It remains to determine tan y" and substitute. The stretches alter tan y into 
tan y', such that 
, cr n /l — r 3 
tan y =-—-tan y. 
Further, by the slide 
cot y" = cot y' — cot D = - .°~ 2 A cot y — cot D. 
/v /v (T lV /1 — r~ 
Hence we have by (lviii.) above 
— cot 
Ej 7 T 
N 2 
cot 
, y/( — cot X cot D ~ cot2 H — 1), 
< - r iV 1 T“ ' V Vo-iv/ 1 — ? / 
/ TT l-TT*\ ^T- ton A/ 
.(1^.)- 
CTo 
or, 
/Ej 7r\ _ o-j tan y 
— tan M - = cot D — — 
\ N 2 / o sm D 
Now the excess E x is the difference of the frequencies in the sum of the strips of 
the volume made by planes parallel to the plane yz on the two sides of the plane ABz 
(defined by y), taken without regard to sign. For on one side of the mean yy this is 
n 2 — n x , and on the other — (w 1 — n 2 ). Hence we have this definition of E L , the 
column excess for any line through the mean of a correlation table : Add up the 
frequencies above and below the line in each column and take their differences without 
regard to sign , and their sum is the column excess. 
If we are dealing with an actual correlation table and not with a method ot 
collecting statistics, then care must be taken to properly proportion the frequencies 
in the column in which the mean occurs, and also in the groups which are crossed by 
the line. It is the difficulty of doing this satisfactorily, especially if the grouping, as 
in eye and coat colour, is large and somewhat rough, that hinders the effective use of 
the method, if the statistics have not been collected ad hoc. 
Now let Eo be the row excess for the line AB, defined in like manner, then we have 
in the same way 
