533] 
FACTORIALS AND DERIVATIONS. 
469 
where the symbols C denote the combinations of the annexed letters taken together 
according to the suffix number; the last coefficient C 5 (a, b, c, d, e) is of course 
= abcde. This is the ordinary theorem giving the expression of an equation in terms of 
its roots. 
Combining the two theorems, if in the first theorem we express the products 
(h — a) (h — /3) (h — 7) (h — ¿5) (h — e), «fee. in powers of h by means of the second theorem ; 
or if in the second theorem we express the powers h 5 , Id, «See. in terms of the products 
(h — a) (h — b) (h — c) (h — d) (h — e), «fee. by means of the first theorem ; then in either 
case we obtain certain identical relations connecting the C, H of (a, /3, ...) or of 
(a, b, ...). 
I have mentioned the factorial notation 
[ni\ r = m (m — 1) ... (m — r + 1), 
where r is a positive integer; a consequence of this is 
[ m ]r+a _ _ r ]q 
where r and s are positive integers ; or as this may also be written 
[m H- r\ r+s = [m + r] r [m] s . 
Assuming this to subsist for s = 0, or a negative integer; first for s = 0, we have 
[m + rf = [in + r] r [m] (> ; that is, [w]° is = 1; and then for s — — r, we have 1 = [m+r] r [ni\~ r ; 
that is, 
and in particular, r = 1, 2, «fee., we have 
[m] -1 
1 
m +1 ’ 
(m + 1) (m + 2) ’ 
«fee., 
which explains the extension of the factorial notation to negative integer values of the 
index. 
But the equation 
[ m ]r+s _ [ m Y 
does not in any determinate manner lead to an extension of the factorial notation to 
fractional or other values of the index. In fact, assuming [m] r = jjjm~~r) ’ w ^ ere ^ 
is an arbitrary functional symbol, the equation in question becomes 
II m _ IIm IT (m — r) 
II (m — r — s) II (m — r) II (m — r — s) ’ 
viz. the original equation is identically satisfied, without any condition whatever being 
imposed upon the function II, and on this account we have not, in the notation of the 
factorial with an integer index, any sufficient basis for a theory of general differentiation.