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of Edinburgh , Session 1878-79. 
mother to her sons, and (4) of a mother to her daughters. These 
may he denoted respectively by 
s , d , cr , 8 . 
7. Superior indices and inferior indices. In an investigation 
we may require to consider several different s relationships ; these 
are best distinguished by means of superior indices, e.y., 
s\ s 2 , s 3 . 
We also require to consider one of the s’s, say the nth.) this is pro- 
perly denoted by s n . 
8. The sign = . 
Let 
sA = B + C + D. 
This equation asserts that the sons of A comprise B, C, and D 
exactly, neither more nor fewer. 
The equation sA = o-B 
denotes that the sons of the man A are identical with the sons of 
the woman B. The truth of the equation involves 
s = cr , 
where the bar denotes that the arithmetical value of the symbol is 
taken. What = denotes is identity of persons — not necessarily 
identity of relationships. Equations which express the latter idea 
may be called identities. 
9. The sign + has its ordinary meaning. Eor example — 
sA = o^B + cr 2 C 
The sons of the man A are identical with the sons of the woman B, 
together with the sons of the woman C. 
10. The law s + d = d + s . 
It is obvious that the formal law 
(s + d) A = (d + s) A, 
which means — 
The sons together with the daughters of A are identical with 
the daughters together with the sons of A — 
is true whoever A be. 
11. The sign x. 
s\ A 
denotes the rath son of the rath son of the man A. The operators 
