242 ON THE EQUATION OF DIFFERENCES FOR AN EQUATION OF ANY ORDER. [262 
The equations successively considered are 
{a, b, c \v, l) 2 = 0, 
{a, b, c, d l[v, l) 3 = 0, 
(a, b, c, d, e l) 4 = 0, 
(a, b, c, d, e, f\v, l) 5 = 0. 
The equation of differences for the quadric, and that for the squares of the roots, 
are considered to be known, and the other results are derived from them : it will be 
convenient to write down in the first instance the results for the quadric, the cubic, 
and the quartic equations, and then explain the process of obtaining them. 
For the quadric equation, 
Equation of differences is, 0 = 
a 2 x 
+ 1 
ac + 4 
b 2 - 1 
Equation for the squares of the roots is, 0 = 
( 
a 2 x 
r- > 
+1 
etc + 2 
b 2 -1 
c 2 + 1 
l) 2 - 
For the cubic equation, 
Equation of differences is, 0 = 
a 4 x 
à 2 x 
r N 
+ 1 
ac + 6 
b 2 - 2 
a 2 c 2 + 9 
ab 2 c — 6 
b 4 + 1 
d-æ + 27 
abed — 18 
ac 3 + 4 
b 3 d +4 
b 2 c 2 -1 
Equation for the squares of the roots is, 0 = 
a 2 x 
S A N 
+ 1 
ac + 2 
c 2 +1 
cP + l 
b 2 -1 
bd- 2 
( 
P, I) 1 -