2 THEORY OF STATISTICS. 
with the subscripts 34 . . . . (n—1) added throughout. The 
coefficient by,  , may therefore be regarded as determined 
from a regression-equation of the form 
Trap le in=1)= IRE oh Diniz. m—=1) + Tn34... (n—1) 
Z.e. it is the ‘partial regression of #,,, .. OD gy .... mp 
Tn3 ....m-y Deing given. As any other secondary suffix might 
have been eliminated in lieu of », we might also regard it as 
the partial regression of 145, .. OD %ay5. .. ny %sa5....n DEINE 
given, and so on. 
13. From equation (11) we may readily obtain a corresponding 
equation for correlations. For (11) may be written 
Brits i Trot... 0-1)” Toe ceo tn=) Tse... (0-1) TL... (n-1). 
Sli i 0234. ... (1) 
Hence, writing down the corresponding expression for bys...» 
and taking the square root 
Test....n=1) —Tmnss.... 0-1) Ton3s....(n-1) 
Ios ein ei te Rt sR L2 
ia EEE DT az 
This is, similarly, the expression for three variables 
rama Tan Lyn Tl 
2 EY -n) 
with the secondary subscripts added throughout, and ry, 
can be assigned interpretations corresponding to those of 6,53, 
above. Evidently equation (12) permits of an absolute check or 
the arithmetic in the calculation of all partial coefficients of an 
order higher than the first, for any one of the secondary suffixes 
of 7153... . » can be eliminated so as to obtain another equation of 
the same form as (12), and the value obtained for 7,5  ..  , by 
inserting the values of the coefficients of lower order in the 
expression on the right must be the same in each case. 
14. The equations now obtained provide all that is necessary 
for the arithmetical solution of problems in multiple correlation. 
The best mode of procedure on the whole, having calculated all 
the correlations and standard-deviations of order zero, is (1) to 
calculate the correlations of higher order by successive applications 
of equation (12); (2) to calculate any required standard-deviations 
by equation (10); (3) to calculate any required regressions by 
equation (8): the use of equation (11) for calculating the 
regressions of successive orders directly from each other is com- 
paratively clumsy. We will give two illustrations, the first for 
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