2 8 Dt\ Allman's Methods of clearing Equations of 
any one of these quantities, as the unknown quantity of 
a cubic equation : arranged it may stand thus, 
= 0, an equation wanting the second term. Therefore, con- 
versely, X =■ — y — 2; ; or else, by dividing the cubic equa- 
tion by the simple one x = 0, and solving the quote, the 
_L 
quadratic equation, x' x —yz = 0, x = ± s / — 3. 
■\-z^ 
Then, in any cubic equation wanting the second term, v. g. 
x^^ qx r — 0, suppose, — ^yz = q ; and, y^ z^ = r: 
then, 2; = — — ; z^ = —y and — ~~ = r : therefore, 
=zry^ -{- — y^ = -^r ± \/ ^ and, by subtracts 
27 27 
ing this from, y-h = = \/^ + therefore,. 
— y — %, ox x=\/— 
\^—\r — 
If the cubic equation have but one possible root, ± 
V —S will represent the two impossible roots. If the cubic 
equation have all its roots possible, the last named expression, 
as well as — y — z, implies the extraction of the cube root of 
an impossible binomial ; except, however, in this single case, 
when two of the roots of the cubic equation are equal to each 
other: then, the solution by the above rule is possible, though 
all the roots be possible; for then,y = z; and the expres- 
sions and J r'" -f- ~ q^, both vanish : then x^ ^ — ^y‘x -|- 
sy = 0, and the values of x, are — sy, y, and y. 
Since any number, either of quadratic, or of cubic surds. 
