EQUATION. 
9 = 0, then the equation becomes « 3 -f- 
53 _j_ r = o , also, since Sab — q = 0,b = 
5-, and by substitution, a 3 -j- + r = 0, 
3 a &r u 
Q 3 
or — 0, an equation of a 
quadratic form; and by completing the 
square, a 6 -f r a 3 -f - = j - |^ , and « 3 + 
r 
and the values of the cube root of b\ are b 9 
2 2 
- = + v/ — — ; therefore a 3 = — 
2 - 11 ^ 4 27 
Hence, it appears, that there are nine 
values of a + b, three only of which can 
answer the conditions of the equation, the 
others having been introduced by involu- 
tion. These nine values are, 
1. a -f- h. 
2. «+~ 1 +^ / — 3 t. 
27 
., and a : 
3. a -j- 
- \/ — 3 
b. 
•y- L 2±V^~£r Also, since « 
+ 6> + r = 0, 1 ’s-j'Tv/ J 
fore a; = a -f- 6 = \/ r i -j— 
4. 
J+V / ' 
’a-f 6 . 
r 
27’ 
5. =1.+ * / - 3 b + 
Q 1 
21 
4 27 
ry 1 3 1 7 
7 - r — a + 6 - 
4. */ _Inr */£_?!. 
+ V 2 + V 4 27 
We may observe that when the sign of 
n/ — JL, in one part of the expression, 
is positive, it is negative in the other, that 
»*=V / -t+v/$-£ + 
Since b = — , the value of x is also 
o u 
2 
- 1 -V' 
-b. 
U. 
* — -/ 3 “+ 
In the operation we assume 3 a b = 9, 
that is, the product of the corresponding 
values of a and 6 is supposed to be possible. 
This consideration excludes the 2 d ‘ 3 d- 4 th ■ 
5th. 7*. an( j 9th. values of a -J- b, or, a:; 
therefore the three roots of the equation are 
a+b> — — «h y — bi 
Cor. 3. This solution only extends to those 
cases in which the cubic has two impossi- 
ble roots. 
For if the roots be m -j- a/ 3?q m — a/ 3 n, 
and — 2 m, then — q (the sum of the pro- 
ducts of every two with their signs changed) 
— — 3 m 1 — Sn, and | = m 2 -f- n ; also, r 
(the product of all the roots with their signs 
changed) =2 m 2 — 6 mn, and ~ = to 3 — 
At 
3mn\ and by involution, 
- 6 m* n -}- 9 m 2 n 2 
~ = m 6 -|~ 3 m 2 n-{-3 m? n 2 -f- n 2 — »’ 
2 3 
Hence, r — — — 9 ni 4 » -j- C j»V — » 3 ss 
4 Z( 
