288 
Mr. Wilson’s Essay on the 
ist. (.r 3 — qx -f r = o) the given equation. 
2d. (.r 3 — qx — r = o) that of its roots summed in pairs. 
3d- -z 3 — + 
+ ^ 2 7 ^ = °1 1 
4th. x 3 — 3 qx — v/49 3 — 27^ = o J 
formed by dividing ( — 6 qx* -f 9 ^ x 1 — 4 q 3 -f 27 r 1 = o), 
the equation of the differences, into two wanting the second term. 
22. Now, leaving these considerations for a moment, let us 
speculate upon the further reduction of the equation. If, instead 
of the present form ( x 3 — qx -j- r = o), ( q ) could be supposed 
to vanish as well as (/>), a still more powerful additional rela- 
tion would be given the roots ; for, the equation being then a 
pure cubic, [x z = ri=r), its roots would obviously be the cube 
roots of (r), and all cube roots are alike. If (r) be a cube, 
and ( */r ) be one of its roots, the 
-two similar equations. 
( — — ^ - — - x v^r and — -■ ~ — - x J/r), let (r) be any quan- 
tity whatsoever, real or imaginary. But it is clear, from what 
has been before observed in Art. g, that this reduction is not 
generally possible, since it supposes two contiguous interme- 
diate terms to vanish together, which real roots do not admit 
of : it must therefore be effected by means of some imaginary 
assumption. Those who are conversant in the use of impos- 
sible quantities, will at once perceive, that the addition or sub- 
traction (which in surd quantities is always the same thing, as 
they are equivocal in sign,) of the imaginary surd [s/ y-J 
to each root of the equation, will infallibly cause ( q ) to vanish, 
but the new roots f ^ ^ — a — b-\-^/ — \q) 
so formed, would not have their sum equal to nothing; and 
