no 
im. G. UDNY YULE ON THE THEORY OF CONSISTENCE 
model. The contours shown in this figure and in 
do not at present concern us. 
the subsequent figs. 26, 46, and 4c, 
The hounding planes are 
X y -\- z = O'b 
(a) 
X y — z = O'b 
X — y z = O’b 
(y) 
— .a: -p y -f c = 0-5 
(§)• 
If the ordinate 2 corresponding to given values of x and y be drawn, it will in 
general cut the surface in two points. These determine the upper and lower limits 
to values of 2 consistent with the given values 
of X and y. If however x and y 
determine a point on the plan of one of the edges of the tetrahedron, 2 only cuts 
the surface in one point, and its value can lie inferred. Thus we have the following 
cases of logical inference :— 
Given. 
Inferred. 
a: = 0 
y = 0‘5 — 2 
a = 0-5 
y = 2 
y = 0 
z = O'b — X 
y — O'b 
Z = X 
2 = 0 
11 
0 
Cn 
1 
2 = 0-5 
X = y. 
Each of these cases corresponds, it will be noticed, to a singde-infinity of sj^ecial 
inferences from two data—one to every point on each edge. It is the onl}- instance 
in which six such infinite series of possible inferences occur, that is, six series of exact 
inferences—inferences of a “ universal affirmative ” or “ universal negative,” to use 
the loo'ical terms. 
O 
§ 17. The limits to x. y, or 2 given for this case b}^ equations (a)-(S) at the 
beginning of last section, are precisely those deducihle from quite different considera¬ 
tions for the quadrant frequencies (a1)ove and below average) in the case of normal 
correlation. In that case, if r^g, r^g be the three correlation coefficients, we have 
for the limits to lUg^ 
rh \/l + ^’12"^’13' 
o 
But Ijy the theorem due to Mr. W. F. SHEPrAEDf 
= — cos ff.rTT 
^■13 = - cos 2^77 
I’og = — cos 2277. 
* ‘ Ro}". Soc. Proc.,’ 1897, vol. 60, p. 486. In the first line of the table on p. 486, for “0” on the right 
read ±1. A similar correction is to he made in the ‘ Jonrnal of tlie Roy. Stat. Soc.,’ vol. 60, p. 834. 
t ‘ Phil. Trans.,’ A, 1898, vol. 162, p. 101. 
