ii4 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 = ^±^ ; 

 write T7~r^ for *> 777- for z : then 



p $r/jt.v -f- /tt 3 



whence thefe two equations are formed. 



p — ^rpv -+• ^> 3 



<7 — 7'v 3 + jy»*v, 

 from which it is manifeft, that p, is a divifor of />, and v a divi- 

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

 ^, and amongft the divifors of q for a number v, that will fatis- 

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

 two following, 





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

 and q t then will z = ~~ : but if not, we are to conclude that 

 the value of z cannot be expreffed this way. 



Thus, in the laft example r rr 73 /6 , it is manifeft that 73 



admits no divifor but 1 : therefore p zz 1, and on trial I find 

 j» =r 2, which two numbers fatisfy the two equations, and 



therefore z = — jr. 



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

 cafe, 



and 



