114 A NEW SOLUTION 



The method I have followed to find when this can be done, 

 being very fimple and eafy in pradice, I fhall here briefly de- 

 fcribe it. 



We have r = ^^±^ ; 



'^"^^ 7$7' fo""^. 777- fo"^ =2 : then 

 P — B^'^" + ^' , „ 



whence thefe two equations are formed. 



q zz o' + 3^°f, 

 from which it is manifeft, that [/. is a divifor of p, and \i a divi- 

 for of y. I feek then amongfl the divifors of p for a number 

 (t,, and amongfl the divifors of q for a number v, that will fatis- 

 fy the two equations above : or rather, that will fatisfy thefe 

 two following, 



n =■- , 



u,- zz* 



If two fuch numbers are to be found amongfl the divifors of p 



and a, then will 2 zz ~- : but if not, we are to conclude that 



the value of z cannot be expreffed this way. 



Thus, in the lafl example r = -^^, it is manifefl that 73 



admits no divifor but 1 : therefore fjt' — i, and on trial I find 

 !> rz 2, which two numbers fatisfy the two equations, and 



therefore z zz —^. 



16. The fame method applies alfo to Cafe I. For, in this 

 cafe, 



^z — s' 



and 



