22 
PROFESSOR K. PEARSON ON MATHEMATICAL CONTRIBUTIONS 
Probable error of E x = '67449 \/N (1 — E^/N 2 ) . 
„ „ E a = '67449 v/N(l -E//N 2 ) . . . 
Correlation between errors in E, and E 2 = — \f ^ ~ ^ • 
1 2 V (1 + E 1 /N)(l + E 3 /N) 
'67449 sin Dy/D {it — D) 
Probable error in r = 
v/N 
where D = — (cf. Sheppard, loc. cit ., p. 148). 
N 2 
Probable error in ratio cr 1 /cr 2 = 
. 
+ 2 (* — n) 0 — n) tan (!' 1) tan 
_o /El 7p 
+ (1 - I 2 ) tan 
N 0 
/E 2 tP 
In 2, 
/ E 3 7 T 
vN 2y 
. (lxiv.). 
• ( lxv -)- 
. (lxvi.). 
. (lxvii.), 
(lxviii.). 
The application of the method here discussed to statistics without quantitative 
scale can now be indicated. If the characters we are dealing with have the same 
scale, although it be unknown, then, if the quantitative order be maintained, i.e., 
individuals arranged in order of lightness or darkness of coat or eye-colour, the 
diagonal line on the table at 45° will remain unchanged, however we may suppose 
parts of the scale to be distorted, for the distortion will be the same at corresponding 
points of both axes. Further, if we suppose the mean of the two characters to be the 
same, this 45° line will pass through that mean, and will serve for the line AB of the 
above investigation. In this case w r e must take tan y = 1, and consequently (lxi.) 
becomes 
, /E 2 it 
o-i/o - 3 -- cos ( ~ - 
/ / E i 11 
Hi 
(tax.). 
We can even, when the mean is a considerable way off the 45° line, get, in some 
cases, good results. Thus, the correlation in stature of husband and wife worked out 
by the ordinary product moment process is '2872. But in this case E x = 382*062 
Eo = 806'425, and this gives the correlation '2994. On the other hand, the actual 
ratio of variabilities is 1'12, wBile (lxix.) makes it 2'76 ! This arises from the fact 
that the errors in E : and E 0 , due to the mean being off the 45° line, tend to cancel in 
E, + E 2 , but tend in directly opposite directions in the ratio of the cosines. Similarly 
the correlation between father and son works out ‘5666, which may be compared with 
the values given in Illustration V. below, ranging from '5198 to '5939. Again, 
correlation in eye-colour between husband and wife came out by the excess process 
‘0986, and by the process given earlier in the present Memoir T002. But all these are 
favourable examples, and many others gave much worse results. We ought really only 
to apply it to find cr l /o -. 2 when the means are on the 45° line, as in the correlation of the 
