8 
Proceedings of the Royal Society 
Table III. are all of the third order. The classes in one row may 
be said to form one genus , and each class itself to be a species. Let 
c and y be considered equivalent, then the index of each term in one 
row is the same ; this is the generic property. The specific property 
depends on the differences expressed by c and y. The varieties of 
a species are obtained by introducing the distinction of gender at 
any place where the form of expression for the species has left it 
unrestricted. For example, c 3 A admits of two varieties, namely 
sc 1 A and dc 2 A — the sons of the sons of the sons of A , and the 
daughters of the sons of the sons of A. 
The notions for the different genera of the third order are exhibited 
in the side column. That for the first genus is great grandchildren; 
that for the second is, in its widest extent, grandchildren of parents, 
but if the cases in which the species reduce to species of the first 
order be removed, then the genus-notion is that of nepheios and 
nieces. The notion of the third genus is children of parents of 
children. The enclosed species, which are affected by containing 
c~ 1+1 or y ~ 1+1 express only children ; the other species express step- 
children or children. Similarly for the other genera. 
Another notion useful to consider and for which a name is required, 
is the number of generations between the individuals represented by 
a term and the origin of the term. For example, in c 3 A the 
individuals represented by c 3 A are removed by three generations 
from A ; in c 2 ~ l A by only one generation. This number may be 
called the interval of the relationship. It is the sum of the indices 
of the term, and is constant for all the species of one genus. 
Relationships may also be classified according to the direct or 
inverse form of the first symbol, and the subsequent number of 
changes from the one to the other. For instance, in c 2 ~ 2 A we have 
the first symbol direct, and only one subsequent change. Here A 
and the individuals represented by the term, have a common 
ancestor; in c~ 2+2 A they have a common descendant. In the 
expressions for the relationships, let c and y be considered 
equivalent, and the indices summed in accordance ; then by 
neglecting the numbers but retaining the signs of the index, we 
shall obtain an expression for the quality we are considering, which 
may be called the sign of the relationship. The sign .may begin 
with either + or - , and may end with either, but it must have 
