Of CUBIC EQUATIONS. ici 



4. In order to give to the two equations, investigated above, 

 the utmoft generality of which they are capable, I write £ for 



r, and they finally become, 



I. Rz 3 - — 3r% 2 — 3R2 + r i o, 



II. Rz 3 — 3rz 2 -f 3R2; — - r ~ o, 



in which two equations, R and r reprefent any numbers, poli- 

 tive or negative, and altogether unlimited and arbitrary as to 

 magnitude, 



I consider the two preceding equations as the fimple cafes, 

 or fimple forms, of the two fpecies of cubic equations : And 

 the method of refolution I have to propofe is, to reduce eve- 

 ry cubic equation whatfoever to one or other of thefe two 

 forms. 



The firft of the above forms is an equation belonging to 

 the circle. It exprefTes the relation between the tangent of 

 an arch, and the tangent of the third part of that arch : and it 

 has, in all cafes, three real roots. If we take the angle <p, of 



which the tangent is *?, the radius being unity ; the three roots 

 of the equation, or the three values of 2, are, tan ^, 



tan (-+120 ), and tan (|-f 240 ). 



The fecond of thefe forms is an equation belonging to 

 the hyperbola. It exprefTes the relation between the tan- 

 gents of two hyperbolic fecliors, of which the one is triple of 

 the other ; and it has, in all cafes, only one real root. From 

 the exprefhon affumed (Art. 3.) above, whence this equation 



was deduced, we get, ^^ = tyfj^'- therefore, 



3/1 — T 3/R — T 



- ,3/I- T : ° r > Wntm S R f0r *> * = , 3/R-T : Alld 



fo 2; is found by extracting the cubic root of a given number. 



The 



