402 
Proceedings of Royal Society of Edinburgh. [ sess . 
i 
of tlie important memoirs of his predecessors regarding the functions : 
the twenty-seventh chapter of Garnier’s Analyse Algebrique , to 
which he refers, may very probably indicate the extent of his 
knowledge. 
Premising that he is going to use such symbols as a v a 2 , . . . 
he calls the letter a the “base,” and the complete symbol “an argu- 
ment of the base,” a 1 being the first argument, a 2 the second, and so 
on. Taking then a number of expressions, “ each of which is made 
up of one or more terms, consisting solely of linear arguments of 
different bases, i.e., characters bearing indices below but none above,” 
e.g., the expressions, 
~ b \ , a x — c x ; 
he alters them by writing the index-numbers above , e.p., 
a 1 — b 1 , a 1 - c 1 ; 
takes the product of these resulting expressions in its expanded 
form 
a? - adb 1 - ale 1 + & 1 c 1 ; 
and then reverses the operation on the index-numbers, thus finally 
obtaining 
a 2 -a 1 b 1 ~ a x c x 4- b x c x . 
The full series of these operations he indicates by the letter £, and 
denotes by the name of u zeta-ic multiplication.” Thus, as results 
in zeta-ic multiplication, we have 
£( a i ~ &i)(«i - c i) = «2 ~ a i\ ~ % c i + \c x , 
and £(a L + 6-J 2 = a 2 + 2a x b x + b 2 * 
Further is used to denote that, after the operations £ have been 
performed, the indices are all to • be increased by r, the result of so 
doing being called the zeta-ic product in its r th phase. 
He nexts recalls a notation previously introduced by him for 
the functions which came later to be known shortly as difference- 
products; denoting, for example, 
* 
* He would not even hesitate to extend the use of the symbol, denoting, for 
example, 
a . 2 a 4 
1.2 1. 2.3.4 
1 
. . . by £ cosK). 
