132 
ME. G. UDNY YULE ON THE THEORY OF CONSISTENCE 
the inferior conditions for the groups of third or higher orders only assign these 
extreme values to some, but not all, of the partial coefficients. Thus, for example, 
take the figures of § 32. Applying the inferior conditions of consistence (§ 5,11.) we 
find as limits to (ACD) for the hoys 0 and 123. The remaining frequencies of the 
anaresate must then be— 
0& o 
(ACD) 
. . . . 0 
123 
(ACS) 
.... 142 
19 
(AyD) 
.... 290 
173 
(aCD) 
.... 123 
0 
(ayD) 
.... 370 
493 
(aC8) 
. . , . 20 
143 
(AyS) 
.... 440 
563 
(ayS) 
.... 8609 
8486. 
The aggregate corresponding to the minimum value of (ACD) will evidently give 
minimum values to the partial associations in 'positive universes like j x4.C j D j, but 
maximum values to those in negative universes I AC | S j, AD ] y j, &c. Then inspection 
will show that only for | AC | D | and | AD | C | are the limits ffi 1 ; for the remaining 
associations the limits are— 
|CDIA| 
— 1 
+ -91 
lAC S| 
-h -33 
+ -99 
1 AD|yl 
+ -68 
+ -88 
CDiaj 
— 1 
+ -99. 
In the discussion of the Childhood Society’s material in my previous memoir I 
remarked on the fact that all the partial coefficients of association between defects 
in positive universes were small, while those in negative universes were large and 
positive. I at first thouglit that this might l)e a logical consequence of the given 
values of the second-order grou|js, but this is not the case. The values of the 
associations like |AC|D|, lADjC], &c., are almost indeterminate; certain of the 
coefficients with negative universes (| AC] 81, |AD|y|)are necessarily positive, hut 
others (| CD | a|) may fall to the extreme limit — 1. 
§ 38. It may be useful to remark tliat if two associations [ABj and |x4.Cl, in a 
third-order congruence, are both equal to zero, the limits to the third association 
|BC1 are necessarily 1, whatever the values of po, Pa- If we write pj po for 
(AB)/(U) or X, P 3 for (AC)/(U) or p, the limits to (BC)/(U) or ^ are 
< (L - 
- Ih) {ih + 
(1) 
< 
Ih ilh + Pi - 1) 
(2) 
> 
Ih + Pi (Pi - Pd 
(3) 
> 
Pi + Pi {Pz - Pi) 
(4). 
