29 THEORY OF STATISTICS. 
which is similarly the equation 
ps Ton + Tins - Tony 
2S =v ld = 72a) 
with the secondary suffixes 34 .... (n— 1) added throughout. 
18. Equations (12) and (17) imply that certain limiting 
inequalities must hold between the correlation-coefficients in 
the ‘expression on the right in each case in order that real 
values (values between +1) may be obtained for the correlation- 
coefficient on the left. These inequalities correspond precisely 
with those ‘conditions of consistence” between class-frequencies 
with which we dealt in Chapter II., but we propose to treat them 
only briefly here. Writing (12) in its simplest form for 7, 
we must have 7},,<1 or 
(P19 = 715+ Tos)” 
Et Su] 
(1 = ri) (L — 73) 
that is, 
Tio + 1s + 13 — 2rigrigryy <1 (18) 
if the three 7”s are consistent with each other. If we take ry, 7; 
as known, this gives as limits for 7,, 
Tihs t J1- Th — 7” + riers. 
Similarly writing (17) in its simplest form for 7, in terms of 
Ti9.3 T1500 a0d 755, We must have 
Tas + 155 + 1351 + 2p a1 10 <1 . (19) 
and therefore, if 7,, and r,;, are given, 7,,; must lie between 
the limits 
~ Tipglise WT- Ths — Tigo + TioaTis. 0 
The following table gives the limits of the third coefficient in 
a few special cases, for the three coefficients of zero order and 
of the first order respectively :(— 
Value of Limits of 
712 OT 712.3. | 713 OT 713.2. 723. 723.1. 
0 - 0 Fl +1 
+1 == +1 | = 
x1 Ti = TI +1 
+4/0°5 +1/0°5 0 LIER) 
+705 FA/0'5 00-1 0, +1 
“Hy