V. A New Method of refolving Cubic Equations. By 

 James Ivory, Esq^ Communicated by John Playfair, 

 F.R.S. Edin. and Prof e [for of Mathematics in the Univerfity 

 of Edinburgh. 



[Read, 6th May 1799.] 



1. [" Divide cubic equations into two varieties or fpecies : the 

 JL one, comprehending all cubic equations with three real 

 roots ; the other, all thofe with only one real root. 



2. Let <p denote any angle whatever, and let r zz tan <p, the 



radius being unity : let alfo z =: tan - : then from the doctrine 



of angular fe&ions we have 



3 g — * % 



r _ 



3*" 



which being reduced to the form of an equation, is 

 z 3 — $rz x — 32 -}- r zz o. 

 Now, from what is commonly taught in angular feclions, z, 

 in this equation, may denote, not only tan -, but alfo 



tan(!+i2o°J, or tan (- + 240°). It is to be remarked, 



too, that any value whatfoever may be ailigned to r, pofitive or 

 negative, and without limit or reflriction as to magnitude. The 

 equation, then, has three different values of z for every given 

 value of r ; and it belongs to the fpecies of cubic equations, ha- 

 ving three real roots. 



N 2 3. Again 



