124 
ME. G. UDNY YULE OX THE THEOEY OF COXSISTEXCE 
It must 1:)e noted that in the type of cases at present under discussion q cannot be 
less than -^th, or OH66 . . . , for, hy the i'A/rd-degree conditions of consistence, 
(AB) + (AC) + (BO) < (A) + (B) + (C) - (U) 
that is it 
(AB) = (AC) = (BC) = q . (U) 
(A) = (B) = (C) = A . (U) 
3p <1 q< i 
For this value of q the plaiies (a) and (e) fall together into the origin ; the whole 
surface, so to sj^eak, closes up. 
§ 30. Case (3)— 
(AB)/(U) = (BC)/(U) = &c. = 0-3. 
In this case the pairs of attributes are all positively associated. The first case is 
therefore intermediate between the second case and the present (')ne. 
I’he })lanes representing the conditions of inferior congruence are 
X = O'1 y = O'I z — O'l 
X = 0'3 y — 0‘3 2 = 0’3, 
hut, as in Case (l), these do not actually come into account, only passing through the 
edges AC, BP, CD, and FM of the octahedron bounded Ijy planes 
X y z = O'-i (a) 
X dr y — z = X — y -f- s = — .a* -p ?/ -p s = 0'3 {(i, y, S) 
;r + ?y = 2 = 0'8 (e) 
,r + ;?/ — 2 = .r — y + 2 = — x + y + 2 = 0 -L (f, y, 0 ). 
The surface is re})resented in fig. 12 below. To facilitate identification, hut avoid 
)vercrowding the small-scale figui'e with lettering, the planes may be stated to he 
ABC e . . . EDF. 
. . PQBCB. 
. . PBAHG. 
. . CKMHA. 
inferences corresponding to the 
rx 
/3 
. . GDFMH 
. . QIUIFE 
. . PQEDG 
There are eighteen infinite series of definite 
O 
y 
8 
P 
V 
0 
eighteen edges, all reducible to oiie or other of tlie tvpes 
Given. 
± {x - y) = 0-1 
,r = 0'3 
X -P y = 0‘25 
X + y = 0’55 
Inferred. 
2 = 0-2 
2 = y 
z — ()'15 
2 = 0'25. 
