[ 74 ] 



If the roots of a cubic equation are encreafed by 7 the fccond 



term will vanifh, if by — T + \^ q T ^^^ ^^'^^^ t^'""^ "^"^'''^ vanifh, 



therefore if/' = 3 fthe fecond and third terms may be ext-erminated 

 together, therefore the equation will have two impollible roots; 

 hence it appears that the equation of the required form has two 

 impoflible roots,* confequently the value of f, by which the roots are 

 to be encreafed will be impoflible when all the roots of the propofed 

 equation are real •.• <7, whofe cubic root mufl be computed, wjll be 

 an impoflible ^binomial, unlefs in the particular cafes where the 

 coefficient of the impoflible part vaniflies. 



It remains to be proved that when the propofed equation has 

 but one poffible root, the value of e, by which the roots are to be 

 encreafed (or diminiflied) will be pofliblc and confequently a, 



Let the roots be —?«-i- / — «j —m — n/ — n^ —b 



p ^ 2 ?n ■{■ b, q = m- -\- n + 2 b m, r = bm* ■}■ bn 



' . e' . e = 



-39 -9'" -3/"" 



m- 



* That the equation of the required form has two impoflible roots, appears alfo 

 from tills, that two of the cubic roots of a, are impoflible. 



