ioo A NEW SOLUTION 



3. Again I affume this expreflion, 



I — T (1 — 55) 5 



Let there be conceived an equilateral hyperbola, of which 

 the femiaxes are each equal to unity, and let a ftraight line be 

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

 ftraight line to be drawn from the centre, to cut off a feclor 

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

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

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

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

 fhall be one third part of the former feclor, and to intercept a 

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

 as in the expreflion here afTumed, viz. 



1 — T (1 — <z,y 



i-P - (i-f *)'* 



I shall not flop to demonftrate this propofition refpecling 

 the hyperbola : it eafily follows from the known properties of 

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

 ftricl analogy that fubfifts between the two varieties of cubic 

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

 indeed very evident from the nature of the afTumed expreflion, 

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

 correfpondent value, and only one. 



From our afTumed expreflion we get 



_ P-i-O 3 — C 1 — ^) 3 _ 3«4- gl v» 



which being reduced to the form of an equation, is 



Z 3 — 3«« a +32 — r — O. 



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

 -of t ; and it belongs to the fpecies of cubic equations having 



only one real root. 



4. In 



