of Edinburgh, Session 1880-81. 
5 
that case it is convenient to write them on a joined ribbon of paper, 
which may be passed over and under the page on which we are 
working ; in this way the repeated writing of the remainders is 
saved. The saving thereby is considerable ; thus in the computa- 
tion of the tables for converting solar into sidereal time, and 
conversely, the writing of eighty-six thousand figures was spared. 
The above example sufficiently explains the principles and the 
application of the artifice. 
3. Algebra of Kelationship. — Part II. By A. Macfarlane, 
M.A., D.Sc., F.R.S.E. 
(Received October 19, read December 6, 1880.) 
At the end of my previous paper on the Algebra of Relationship 
(Proc. Roy. Soc. Edinb., vol. x. p. 224), I promised to return 
to the investigation of the subject, as it seemed capable of further 
extension. This anticipation has proved correct, and I have now 
the honour of bringing before the Society the further developments 
I have made. 
In this investigation we consider a particular class of objects, and 
that class is in its widest extent mankind, by which term I mean 
the entire number of men who have existed, exist, or will exist. 
The universal properties of the symbols are deduced from the 
universal properties of mankind. In our consideration of this 
universe of mankind, we restrict our attention to the classes into 
which it is divided by qualities depending on ties of anterior or of 
posterior relationship, that is, of consanguinity or of affinity. 
The italic capitals A, B, C , &c., are used to denote the names 
of the individuals within this universe. They are singular terms, or, 
in other words, selective symbols which operating within or upon 
U, select each only one individual. They are subject to the laws 
A + B = B + A; 
A 2 = 0 ; 
and A B = 0, unless B- A ; 
where the sign = means identical with. 
In my previous paper, I based the investigation on four funda- 
mental symbols s, d, cr, 8, denoting respectively sons of a man , 
