226 Proceedings of the Royal Society 
s 1 and s 2 are similar as regards their nature, but are different indi- 
vidually. Consequently tbeir arithmetical values may be different. 
12. In this branch of analysis, order is essential. The operator 
s 2 is prior to s 1 . Hence the commutative law does not apply to 
symbols connected by x . 
s\ n s 2 n is not = s 2 n s 1 tn . 
In this respect the branch of analysis which investigates relationship 
stands in contrast to the branch of analysis which investigates 
quality. Tor in the latter 
xy = yx. 
At the same time it agrees with the Quaternionic analysis, for 
in it 
pq = qp 
is not true in general. (Tait’s “ Quaternions,” p. 37.) 
13. I may here quote a few sentences from a paper by the late 
Professor Clifford, which throw light on this subject — 
“ There are two sides to the notion of a product. When we say 
2x3 = 6, we may regard the product 6 as a number derived from 
the numbers 2 and 3, by a process in which they play similar parts ; 
or we may regard it as derived from the number 3 by the operation 
of doubling. In the former view, 2 and 3 are both numbers ; in 
the latter view 3 is a number but 2 is an operator, and the two 
factors play very distinct parts.” The product of two quality 
symbols is of the former kind ; the product of two relationship 
symbols is of the latter kind. 
14. s m A = B 
asserts that B is the mth son of the man A. 
asserts that A is the father of the man B, and that B is the mth 
son. Thus, 
1 _ 
s m 
does not only denote “ father of a male,” but introduces the proper 
number to individualise the relationship. 
