108 
ME. G UDXY YULE OX THE THEOEY OF COXSISTEXCE 
Geometrical Eepeesentation of TifE Conditions of Consistence. 
Congruence of the Ihird Degree. 
§ 15. Since the conditions of consistence for a congruence of the third degree only 
Involve tln’ee second-order frequencies each, it is possible to construct geometrical 
models to represent them, tire first-order terms lieing treated as constants. These 
models exlnbit in such a beautiful manner the nature of the conditions, and the 
limiting character of the cases dealt with in ordinary logic, that it is worth while to 
treat a few special cases at length as illustrations. Tt vdll be convenient to use for 
the present the abbreviated notation 
= (AB)/(U) . y = (AO)/(U) . = (BG)/(U). 
p, = (A)/(U) . p, = (B)/(U) . p, = (C)/(U). 
Then, treating .t, y, z as rectangular co-ordinates, all sets of consistent values ot 
X, y, i must determine points within the space bounded by planes (§ 4 and § 5). 
.X = 0 
or 
X 
— 
Pi 
+ Ih 
1, X = 
Pi 
or X = 
P-2 
y = 0 
?5 
y 
— 
p\ 
+ Pi 
— 
B y = 
Pi 
y = 
Pi 
c = 0 
n 
= 
Ih 
f Pi 
— 
1, - = 
P-2 
1 ? 
Pi 
X 
+ 
y 
+ 
z 
2h 
7h + Pi 
— 
1 (a) 
X 
+ 
y 
— 
= Pi 
(/S) 
X 
— 
y 
+ 
z 
= p-i 
(r) 
— X 
+ 
y 
+ 
z 
= Pi 
(A) . 
(B) . 
It is convenient to regard the equilateral tetrahedron bounded b}^ the four planes 
(«) — (8), representing tlie siq)erior conditions of consistence, as the fundamental 
“congruence surface,” its edges being truncated more or less l)y the planes (A). 
Only six of the planes (A) at most can of course come into account at one time, 
the remaining six lying outside the surface. 
The lines (a/3), (ay), (a8), &c., in vdiich the planes a, (B, y, 8 meet, are all parallel 
to one or other of the co-ordinate planes : thus we have for 
(a/3) 
. = 1 
(/A 
+ Pi - 
I) 
(ay) 
2/ = V 
{Pi 
+ Pi - 
1) 
(a8) 
■x = A 
{Pi 
+ P-z — 
1) 
ih) 
;r = i 
{pi 
+ Pz) 
m 
2/ = i 
{th 
+ /b) 
(yS) 
{P2 
+ Pi) 
(0)- 
