120 
MI!. G. UDNY YULE OX THE THEOEY OE CONSISTENCE 
Since, however, each of the hounding hj^per-planes is parallel to one or other of the 
axes, four surfaces in three dimensions may be substituted for the hyper-surface in 
four dimensions. If we denote 
(ABC)/(7^), (ABD)/(^0, (ACD)/(w), (BCD)/(u), 
by rr, y, w, then the surfaces representing the consistence of [1] xzy, [2] xyw, 
[3] xzw, [4] yziv, respectively, for given values of the frequencies of lower orders, will 
in general differ from each other. If we desire to find the limits to values of w for 
given values of .r, y, z, we must fii’st see thtit the values of x, y, z are consistent from 
surface [1], then find the three pairs of major and minor limits to iv given by 
surfaces [2], [3], and [4]. The lowest of the major limits and highest of the minor 
limits so given are the true limits to w. 
We may take as examples of fourth-degree congruence-surfaces three of the very 
simplest cases, in which (l) equality of contraries subsists for the first and second- 
order frequencies, and (2) actual equality subsists between all the second order 
frequencies, i.e., (AB) = (AC) = (AD) = (BC) = &c. These are highly specialised 
examples, but are of some interest for their bearing on the theory of normal or quasi- 
normal coi'relation. 
It must be remembered that in a normal distribution of frecjuency, where the 
divisions between A and a, B and &c., are taken at the means, the frequencies of 
all orders are definitely determined hy the second-order frequencies, and complete 
ecjuality of contraries subsists for frequencies of all orders. This is not the case 
wdiere the only datum is that equality of contraries subsists for frequencies of the 
first and, tlierefore, also of the second order. The ecjuality of contraries need not, as 
shown in my previous memoir,"" spread to the frequencies of the third order, but if it 
be assumed to do so the latter l)ecome determinate as in the “ normal ” case. 
§ 27. For the simple type of cases considered, the equations to the bounding-planes 
of the congruence-surfaces reduce to the forms given below. The four component 
surfaces for xyz, xyw, &c., are, of course, all identical, so it is only necessary to 
consider one in each case. The first conditions given are tliose of inferior congruence 
[cf. § 5), and for brevity we have written 
(AB)/(«,) = (AC)/(u,) ^ &c. = q 
X = 0 
o 
11 
S =: 0 
= 2q — 
0-5 
= -<2 - 
0-5 
11 
— U-5 
X = q 
y = q 
== 2 
= 3q — 
(J-5 
= 3q - 
0-5 
= 3q 
— O'o 
* “ On the Association of Attributes in Statistics,” ‘ Phil. Trans.,’ A, vol. 194, pp. 263 ei seq. 
