XII.—PARTIAL CORRELATION. 
This is again the relation of the familiar form— 
Ola = ai(1 TT 3.) 
with the secondary suffixes 23 . . . . (n—1) added throughout. 
It is clear from (9) that 7,65... (u_1), like any correlation of order 
zero, cannot be numerically greater than unity. It also follows 
at once that if we have been estimating z, from Zor Ty xv in» puis 
x, will not increase the accuracy of estimate unless 7,0, (n=1) 
(not 7y,) differ from zero. This condition is somewhat interesting, 
as it leads to rather unexpected results. For example, if 7, = + 0-8, 
r3= +04, 793= +05, it will not be possible to estimate #, with 
any greater accuracy from x, and x; than from z, alone, for the 
value of 74, is zero (see below, § 13). 
11. It should be noted that, in equation (9), any other subscript 
can be eliminated in the same way as subscript » from the suffix of 
Oss. ...n SO that a standard-deviation of order p can be expressed 
in » ways in terms of standard-deviations of the next lower order. 
This is useful as affording an independent check on arithmetic. 
Further, 0,9; (ay can be expressed in the same way in terms 
of i935... (ng, and so on, so that we must have 
iss...n=0(1 -7H)(1 - 715)(1 = TTazs) + «© (1- Tinss...m-1) « (10) 
This is an extremely convenient expression for arithmetical use ; 
the arithmetic can again be subjected to an absolute check by 
eliminating the subscripts in a different, say the inverse, order. 
Apart from the algebraic proof, it is obvious that the values must 
be identical ; for if we are estimating one variable from = others, it 
is clearly indifferent in what order the latter are taken into account. 
12. Any regression of order » may be expressed in terms of 
regressions of order p — 1. For we have 
2(21.84... m0 22.34... 0) = (1.34... (n—=1) 234... m) 
=2a154... (n-1)(T2— b2n.34... (n-1) . Tn — terms in 24 to 2p) 
=2(x1.34... (0-1) 22.34... (n—1)) = bon.3s... n-1)Z(X1.34... n=1)+ Tn34. .. m=1)). 
Replacing b,, | m=) DY Ons (n—1) * 03s... 1/0 vo. (n=1) 
we have 
b12.31...n. 0534... n=b1a34... (n=1)+ 05.34 . .. (n—1) —O1n.34. .. n—1). bn284 .(n—1). C334. . . (n=1), 
or, from (9), 
b -b b 
b = 1234 .... (n—-1) 1n34....(n-1)* Yn2.34 «ae (n=1) 1} 
8 1 =Buss.... 000 Durst so toed) £1 
The student should note that this is an expression of the form 
b br J b1n - Ons 
12.n 1 T ban L bs 
237