10 
Proceedings of the Royal Society 
moral laws are observed. For example, cAyS^A and cAy 2 ~ l A are 
necessarily non-existent where the Christian laws of marriage are 
observed. 
As an example of the manner in which the notation of this paper 
may be used, I shall employ it to find the different compound 
genera species and varieties of the term cousin. The word cousin 
in its general sense means any relationship of the sign m - n where 
m and n is each not less than 2. When both of them are two, we 
have the relationship cousin in its strictest sense ; when both are 
3 we have second cousin, and I suppose that 2-3 and 3-2 have 
to be expressed by means of the same phrase. But by putting in 
the particular numbers for m and n , we obtain a simple and perfect 
means of specifying all the possible elementary forms of the notion. 
I shall restrict the elementary form of the word to the form 2-2, 
which coincides with the fourth genus of the fourth order of re- 
lationships (the notion of which is grandchildren of grandparents), 
provided we exclude all the instances in which that genus reduces 
to genera of a lower order. 
The different species are, 
,, 2-2 - 2 ^ 1-1 . 1-1 .-1 r 1-2 v ,. l "2 ^,, 1 - 1-1 v 2 - l.-l 2-2 . 
> 6 7 ’ c y 6 5 c y > y° > y° y •> y 0 ? y > 
and suppose them numbered consecutively from left to right. The 
different combinations of these species two together form the com- 
pound genus of the second degree. The number of species in this 
compound genus is 28, but certain of them do not exist on account 
of the moral laws 
C 2 Ac~ l y~ l A — 0 and y~ 1 c~ 1 Ay~ 2 A = 0, 
or their equivalent forms, 
ycAc s A = 0, aDd y^AcyA = 0. 
The non-existent species are 12, 15, 26, 34, 37, 48, 56, 78. There 
are 20 left. The number of species in the compound genus of the 
third degree is naturally 56, but by the above laws it is reduced to 
16. For that of the fourth degree the numbers are 70 and 4 re- 
spectively. The four are 1368, 1467, 2358, 2457, each of which 
represents what ought to be meant by the phrase a full cousin; for 
all the higher compound genera are non-existent. Each species of 
each of these compound genera may contain four varieties ; for the 
