412 
Biometric Notes 
to investigate the probable error of this coefficient on the lines laid down by Pearson and Filon 
in their classic paper on the probable errors of frequency constants*. 
Let 3pio be the correlation between variables 1 and 2 for a constant value of 3. 
Then 
V ^ - ' 13 V -l - ' 23 
Taking logarithmic differentials, we find, 
3P12 ni-nzr-a (>i2- '-13'23)(l-'-13-) (n2 " '•l3?'23) (1 " »W-^) ' 
Squaring, summing and dividing both sides by n, the total number of observations, and 
substituting for the standard deviation of the errors in ri2, rig, r23, and for the correlations of 
errors in those quantities the values given by Pearson and Filon in the paper already cited, we 
find after striking out the common foctors (1 - (1 - V'r^) 
2^3P12 ('-12 - >-13 ^-23)- ( 1 - ni) ( 1 - T.,i) ^ 
3Pl2- 
= (1 - l\r - - >V32 + 2/-i2 ''13 '23) {(1 - 'W") (''23 - '•l2''l3) r^lTiz 
+ ( 1 - '-is'-) ('-13 - ras) »-i2 '-23 - (^3 - »-12?-23) ('•23 " ?-]2 '"is) >-13'-23} 
+ (1 - r^i) (1 - /W-) {(1 - n^) + ('-13 - n2'-23)' + (»-23 - niTxzf " 2r23(l - v^^) {r^-mn^ 
-2(1- (r,3 - r23) ri3 + 2 (rjg - r<y^ - 7\2 rn) >'i2}- 
But the second part of the expression to the right of the sign of equality reduces to 
( 1 - ni) ( 1 - raaS) ( 1 - r,^) ( 1 - - r^' - r.J + 2r,2 ri,r,s), 
and the second factor of the first part reduces to 
1 - ^2^ - '13^ - + 2/-,2 '-13 ^23 - ( 1 - '-12-) (1 - '-13") (1 - ^•23^), 
SO that the whole expression on the right reduces at once to 
(1 — i'r/ ~ ^l'/ ~ '"23" + 2'/'i2 ''l3''23)^- 
But 
(l-n3^)(l-'23^)' 
so that 
23pl2 = 
^ 1 - 3P1 2" 
and the probable error of spio is 
_ - 67449 (1 -3P12'') 
and is thus, as stated by Yule, of the same form as the probable error of the direct correlation 
between any two variables. 
* "On the Probable Errors of Frequency Constants and on the Influence of Eandom Selection on 
Variation and Correlation." By Karl Pearson, P.K.S., and L. N. G. Filon, B.A., Phil. Trans. R.S., A, 
Vol. 191, pp. 229 et seq. 
