loo A NEW SO LUTION 



3. Ac A IX I affume this expreflion, 



I 



i-f . - 



Let there be conceived an equilateral hyperbola, of which 

 the fcmiaxcs are each equal to unity, and let a ftraight Hne be 

 drawn to touch the hyperbola at its vertex : Conceive alfo a 

 ftraight hne to be drawn from the centre, to cut off a fector 

 from the hyperbola itfelf, or from its oppofite, or conjugate hy- 

 perbolas, and to intercept a part r (eflimated from the vertex) 

 on the tangent line : And, in like manner, let another ftraight 

 line be drawn from the centre to cut off another fedor, that 

 ihall be one third part of the former fedor, and to intercept a 

 part z on the tangent line : Then the relation of t and z will be 

 as in the expreflion here aflumed, viz. 



I— T _ (l— aV 

 x+r - (! + »;)'• 



I SHALL not flop to demenftrate this propofition refpedling 

 the hyperbola : it eafily follows from the known properties of 

 that curve. I mention it merely with the view of marking the 

 ftridl analogy that fubfifts between the two varieties of cubic 

 equations. It is fuflicient for our purpofe to remark, what is 

 indeed very evident from the nature of the aflumed expreflion, 

 that, whatever value be afllgned to r, z has always one real 

 correfpondent value, and only one. 



From our aflumed expreflion we get 



— C' + ^O'— (i— ^)' _ 3H^£_, 

 '■- (.i + ^j' +(»-«)' ~ 1+3^' 

 which being reduced to the form of an equation, is 

 z 5 — 2,tz ^ + 3% — T := o. 

 This equation has only one value of Z' for every given value 

 of r ; and it belongs to the fpecies of cubic equations having 



only one real root. 



4. In 



