The theory of determinants in the historical order of development, by Sir Thomas Muir.

RECURRENTS (HANKEL, 1865) 221
and x = x (ss8sy~+2Sa++y- qa+-8y 2.+ys8)
=1 -r64SS +12sa++y-2 (Sa+y+a+s +S+sa)
3! L -4 (++ysa+534a$ + Sy a+)J
= sass....., as before.
The mode of proof is the gradational, the step being from one value
of n to the next higher.
SIACCI, F. (1865).
[Sull' uso dei determinanti per rappresentare la somma delle potenze
intere dei numeri naturali. Annali di Mat., vii. pp. 19-24.]
Beginning with the determinant
z 1..
z2 1 2.....
z3 1 3 3....
z4 1 4 6... or R(z) say,
' 1 (r)1 (1).... (r)I.-2 (-).Z+ 1 I (r+l)l (- >+1)2 *. *. (r+1)I_2 (0v+l)3r-i
Siacci performs the operation
coll +col2 +z col +z2col- +... +z" -col.,,+,
and obtains at once
ER(z) = R(z+l) — (r+l)!z",
whence R(z+l) = R(z+2) - (r+l)!(z+l),...............
R(z+n) = R(z+nz+1)- (r+l )!(z +)',
and, therefore, by addition and substitution of 0 for z,
R(n+l) = (r+l)! ("+2"+... +n)
as desired.
Siacci, however, goes on to point out that by performing on R(z)
the operation
coll col,2-z col +z2col 4-z3col-...