198 
Transactions of the Boyal Society of South Africa. 
2. In continuation attention is now called to the alternative series of 
identities, 
11^41 = ^3, \{a, a^^\^ a,^\h^ + a.]\ + a.\^\^h^\\, 
a. a^^ 
c^\ c^ + aj C4) 
where the function of the ?^th degree is expressed as the sum of n terms, 
each of which is the product of one of the elements of the first frame-line 
by a function of the next lower degree, and where the first term involves 
all the a's, the second term all the a's except one, the third term all the 
^'s except two, and so on. To construct the development in any case, say 
the case of 
|«2 <^ 
we first delete the a's to obtain the minor which is the basis of the 
second factor of every term. Then we complete this minor for its place 
in the first term by increasing each of its by the corresponding a, for 
its place in the second term by increasing each of the c's in the same way, 
and so on, the result thus being 
^2 1 1 ^3 + a. + a^ + 
+ a.Ub. 
d. . 
(I.) 
^3 ^ 
f/5 + «, 
+ a-. 
"4 ^5 
d. 
whence are obtained 50 + 35 + 24 + 16, i.e., 125 final terms.''' 
* A useful rule-of-thumb for obtaining the development in the case of the function 
of the 3rd order 
1 1 ^2 
2/3 2/4' 
is : Multiply xjz^ hy the sum of the remaining elements, x^y^ by the similar sum, x^y^ by 
the similar sum, and make a term from the elemeyits of each frame-line. 
It is convenient also to use ^ as the notational letter for the function, the degree 
being specified where necessary by a suffix. 
