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 correlationcoefficient
 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

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