FOR AN EQUATION OF ANY ORDER. 
95 
For the quadric equation, 
Equation of differences is, 0= 
a“X 
A 
A 
f ^ 
< ^ 
+1 
+ 4 ae 
-1 ¥ 
1 ). 
Equation for the squares of the roots is, 0 = 
( 
or X 
A 
f ^ 
+ 1 
r 1 
+ 2 ac 
-1 
+ 1 r 
1 )^ 
For the cubic equation. 
Equation of differences is, 0 = 
X 
A 
a^x 
A 
f ^ 
+ 1 
( ^ ^ 
+ 6 ac 
-2 b- 
( 1 
+ 9 «V 
— 6 ab'^c 
+ 1 ¥ 
+ 27 a^d- 
— 18 abed 
+ 4 ac^ 
+ 4 b^d 
— 1 S'c* 
!/• 
Equation for the squares of the roots is, 0= 
( 
fl- X 
A . 
A 
A 
A 
f — ■" >1 
+ 1 
( 'I 
+ 2 ac 
-1 b^ 
f „ '> 
+ 1 c' 
— 2 bd 
( '1 
+ 1 d^ 
1 )^- 
0 2 
