Q90 
Mr. Wilson’s Essay on the 
(•z 3 — 31-2 = 0); to find the similar equation of their diffe- 
rences (a; 3 — 9 r -|- * = °) ; and, to find the equation produced 
by the quantities formed from the addition of the roots of the 
one to those of the other multiplied into the imaginary surd 
(v/ — -§•)• The equation last found would, however, be of the 
dimension of the 9th power, at least : for, the addition of each 
root of the second equation to every separate root of the first, 
produces a separate quantity : thus, 
( 2 ,— i, — 
being the rootsof the 1st. / 
•1 + 0 
■1+0 
(°.+v/-3,-V-3)1 2 +3v/-i--i+3v/-i,-i+3v/-i > 
those of the 2 d. - [2— 3v /— i,— 1— 3v /— i,— 1— 3v /_iJ 
will be the nine quantities formed by their addition. But we 
have a decisive clue to distinguish some from the rest ; for we 
know, that if we find the equation of the cubes of those quan- 
tities, it must have three equal roots ; for, every time the sum of 
two of the roots of the first equation meets its own difference, 
it will constitute a cube root of (8), and therefore, the equation 
(a; 3 — 8 = 0) will be three times contained in the resulting 
equation of cubes. That equal root being discovered by the 
method of finding equal roots, so often alluded to before, re- 
duces the equation (a: 3 — 3 a; -j- 2 = o) to the pure cubic 
(*' = 8). 
24. The instance in the last article, of the reduction of the 
equation (a: 3 — 3a: -f- 2 = o) to a pure cubic, by means of 
the equation (a: 3 -f 3a; = o), evidently depends upon the 
coefficient of the second term vanishing ; and also, that of the 
third term being the same in both, but of opposite signs. For, 
the roots of the one, in their combinations by two, producing 
( — 3), and those of the other (-f 3 ), of course destroy each 
