9 
of Edinburgh, Session 1880-81. 
them in alternate succession. The sign of eye is + , of c 2 -1 + - , 
of cy 1 ~ 1 c~ 1 -f - , of c" 1 y 1_1 c - + - and so on. The term 
degree may be used to denote the number of pluses and minuses in 
a sign, and positive to denote that the sign begins with + and 
negative to denote that it begins with minus. 
Relationship terms, inasmuch as they select a number of individuals 
from the universe of mankind in such a way that the same individual 
is selected only once, are qualitative symbols of the kind first dis- 
cussed by Boole, and which I have treated of in my book on the 
“ Algebra of Logic.” They are therefore subject to the laws and pro- 
cesses of that Algebra. 
That Algebra supplies us with the mode of formation of a com- 
pound term. Suppose that cA and y B have some members in 
common ; these are classed together by the expression cAyB , where 
cA and y B stand to each other in the relation of simultaneous 
multiplication, a relation quite different from that existing between 
c and A in cA. We may have compound terms of any degree of 
complexity, and the word degree may be used as meaning the 
number of simple terms in a compound term. The number of 
origins in a compound term is the same as the degree, but some of 
them may be coincident. The different compound terms of the 
second degree, each simple term being of the first order, are the 
following : — 
cBcA 
cBe~ x A 
e~ x BeA 
c-'Bc-'A 
cByA yBcA yByA 
cBy~ 1 A y Bc~ x A y By~ x A 
C~ Y ByA y~ 1 BcA y “ 1 ByA 
C~ 1 By~ 1 A y-'Bcr'A y^By^A 
The permutations cByA and yBcA are not different in form; for 
the order of the components in a compound term is inessential. In 
the case of the extremes of the first row, the origins B and A must 
be identical, else the terms are null. In the case of the middle 
terms of that row the origins must be different, and different in sex. 
In the case of the terms of the second and third rows, the origins 
must be different. In the fourth row the extreme terms may have 
origins the same or different, and the middle terms never exist. 
Thus certain compound terms are non-existent on account of 
natural laws ; there are others which are non-existent, where certain 
VOL. XI. 
B 
