V. A New Method of rejolvhig Cubic Equations. By 

 James Ivory, Esq^ Communicated by John Playfair, 

 F.R.S. Edin. and Profeffor of Mathematics in the Univerfity 

 of Edinburgh. 



[Read, 6th May 1799.} 



I. T Divide cubic equations into two varieties or fpecies : the 

 J. 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 dodrine 



of angular feftions we have 



3 Z — z^ 



■^ — - — 3Z" 



1 — 3z 



which being reduced to the form of an equation, is 

 z 5 — ^rz '■ — 32 + T r: o. 

 Now, from what is commonly taught in angular fedlions, x,. 

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



tan [|+ 120°), or tan ^^ + 240°). It is to be remarked, 



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

 negative, and without limit or reflri(5tion as to magnitude. The 

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

 value of 7 ; and it belongs to the fpecies of cubic equations, ha- 

 ving three real roots. 



N 2 3. Again 



