294 
Mr. Wilson’s Essay on the 
(q = a z +a 6+ b z ) and (4 q 3 — 4 a 6 + 1 2 a s b + 24 a 4 f>*+ 28 a 3 b 3 + 24 a 1 &*+ 1 2 a i s + 46 s ) , 
also ( — r—a 1 b+ a b z ) and (27 r z — - - - - - - - 27 « 4 & l +54tf 3 i> 3 + 27 a 1 b* ), 
whence (49 s — -27 r 1 =4a 6 +i2a 5 b — 3 a*b z — 26 a 3 6 3 — 3 a 1 6 4 + 12 a Z> 5 -j- 4 6* ) 
and its square root, or (*/ \q z — 27 r*j— 2 a 3 + 3 a 1 ^ — 3 a b z — 2 b 3 ). Therefore, the quantity 
j ; 1 1 j TlJ) 1 2 a 3 + 3^b — 3 a b z — 2 ft 3 ] f v / 4o 3 — 27 r 1 
4 *F 6+^F| + ^77— ! l= 4 *)-r + - ^_ 3 7 • 
The equation (a: 3 — 4- r = o) may hence be reduced to a 
pure cubic of this form (x 3 = 4 x j — r 4- — )» which, 
when cleared of its irrational quadratic surd, becomes 
{x*+ 8 rx 3 + 16 r> = or (x 6 + 8rx 3 + 
= 0); or, dividing its roots by (2) to reduce it still lower, 
(jr 6 -f- rx 3 -f- -|^-=o), the common reducing equation ob- 
tained by Cardan’s rule. 
Example 2 d. Let (1, 2, — 3), the roots of the equation 
( x 3 — 7x4-6 = o), be taken; the binomials formed from them 
will be, according to the directions of the rule, 
1 + 2 = 
1—2 = 
if} 3-^-t 
3-W-j 
3 J V — y 
-2-4v/-f 
-2-4%/- f 
-i + S%/- J 
— 1 + 5 %/ — T 
+ I 
04 04 
11 II 
9-6 V- f 
. — T 
4 +i 6 v/— 1 
li 
3 
IO\/— f 
N. B. It is necessary to change^ 
the sign of the middle diffe- 
rence, because their sum must 
26 ^ 
~-6V-\ 
3 — V — f 
-1 +i 6 v/-f 
— 2 - 4%/ — f 
-T-'ov'-T 
- 1+5 V-\ 
+ 1 
04 04 
11 li 
always = o). 
26 — 18 ^— *. 
T -32%/— £ 
t+¥%/- f 
2 2 . 
T+ I°\/ — T 
50 r 1 0 , 
T-— %/_* 
24 -- 3 V- f 
24— - 3 ° y/ | 
Z4--VV- f 
The quantity (24 — - ) is therefore the common cube of 
these 3 binomials. 
