[545 
546] 
535 
ch satisfy 
lere is a 
locus is 
each of 
on to be 
recognise 
b of the 
solution, 
5 account 
pectively. 
THEOEEMS IN 
546. 
RELATION TO CERTAIN SIGN-SYMBOLS. 
[From the Messenger of Mathematics, vol. n. (1873), pp. 17—20.] 
I find the following among my papers: 
Let the latin letters a, h,... denote lines of n signs ±, and the greek letters 
a, /3,... columns of the same number n of signs ±; two symbols of the same kind 
are multiplied together by multiplying their corresponding terms, the product being 
thus a symbol of the same kind; in particular, the product of a symbol by itself, or 
square of a symbol, is a line (or column as the case may be) of +’s: and the symbol 
itself is thus a square root of a line (or column) of +’s. Thus n being = 5, we say 
that the latin letters denote roots of + 4- + + + and the greek letters roots of + 
+ 
+ 
+ 
+. 
The roots a, b, c, d, e will be independent if no one of them is equal to the 
product of all or any of the others; and, this being so, the 32 roots are the terms of 
(1 + d) (1 + 6) (1 + c) (1 -f d) (1 + e); 
it follows that, for any other system of independent roots a', b', c', d', e, we have 
(1 + a!) (1 + b') (1 + c') (1 + d') (1 + e) = (1 + a) (1 + b) (1 + c) (1 + d) (1 + e): 
and conversely if either system be independent and this equation is satisfied, then 
the other system is also independent.