zg 6 Mr. Wilson’s Essay on the 
clear; and the quantities themselves, or, which is the same 
thing, their sums in pairs, be affected with it. However, as 
these latter are to be multiplied by three, that multiplication will 
destroy the fraction when they come again to be multiplied by 
the surd (y/ — i), since (3xv/ — -§- = y/ — 3)* Wherefore, 
the same end, as to reducing the equation, will be obtained, 
-whether, after adding the sums of the roots in pairs to their re- 
spective differences, we multiply the sums by (y/ — 3), or divide 
the differences by it ; as has been already shewn in the 3d 
Example to the last Section of this Article. 
3dly. If any cubic equation wanting the second term, be 
transformed into the equation of that function of its roots, 
formed of the cubes of the binomials arising from joining the 
sum of each pair of roots to its correspondent difference drawn 
into the imaginary fractional surd (y/— ^), or each difference to 
its correspondent sum drawn into the surd (y/~ 3)> the trans- 
formed equation will have among its roots three equal cubes ; 
by finding which, according to the methods of finding equal 
roots, the equation is reduced to a pure cubic. 
4thly. The roots of a cubic equation may be all real ; or 
only one of them real, and the remaining two imaginary. If 
only one be real, they will be of this form ( a, — a±h ^ — 1 j . 
and, by taking their sums and differences according to the rule, 
and multiplying the latter into the(y/ — ■§■), one of the resulting 
binomials will be real, and the other two imaginary : the cube 
produced by them will therefore be real. When all the roots 
are real, if two be equal, one difference necessarily vanishes; 
wherefore, the imaginary factor will only appear about the two 
