162 
Proceedings of the Royal Society 
The following Communications were read : — 
2. On the Intrinsic Muscles of the Mammalian Foot. By 
Dr D. J. Cunningham. 
3. On the Expansion of Rational Fractions, &c. By Mr 
A. H. Anglin. 
4. Algebra of Relationship. Part III. By A. Macfarlane, 
M.A., D.Sc., F.R.S.E. 
§ 1. In my previous papers on this subject* I used the rela- 
tionship terms, not in a representative hut in a class sense ; 
for instance, cA was employed to denote the children of the man 
A, and U was employed to denote the total assemblage of mankind, 
or a limited portion of that assemblage. I have found it useful 
for the purpose on hand to analyse these symbols into their com- 
ponent elements. Let U denote a man representatively, that is, 
any man , then mankind is appropriately represented by % U , where 
has its ordinary mathematical meaning of taking the sum. Also 
U A is the appropriate mathematical expression for the man who has 
the name A, and A standing by itself is to be regarded as a con- 
traction for U A . This notation is useful where, as in the present 
case, the universe of the investigation is composed of individuals ; 
but since the universe may be continuous in its nature, by taking 
U to denote the ivhole , a more general basis is given to the Algebra 
of Logic, and accordingly I adopted that notation in my work on 
the subject. 
§ 2. Similarly, let c denote child in its representative sense, 
then %cA denotes all the children of A \ and with the aid of 
the numerical and certain other symbols we can express one, two, 
three, &c., children of A ; or the only, the two, the three, &c. (as 
the case may be), children of A ; or the first, the second, &c., child 
of A. 
First. IcA, 2cA, 3c^4, &c., may be used to denote a certain child 
* Proc. Koy. Soc. Edinb., vol. x. p. 224, and vol. xi. p. 5. 
