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x^ +px' + ^ X i r = o 

 X' = J' + 3ay' + ^a'y + a^ 

 x=y+li px^= py + 2ptiy+pa^ 



qx = qy + <ja 



r — r 



/>* =9tf' + 6ap + P'' and 3^- = g^'' + 6a/> + 3 7 therefore/"' and 3^. 



cncreafed by the fame quantity, viz. ga' + 6 ap 



x' +px'' + q X + r = 



y 

 }( — — y^ -h pay^ -^ q a' y + a^ r=o 



a 



*' =p^ a'' and ^9—39^' therefore both multiphed by the fame 

 quantity, viz. a"". 



Hence it appears that the problem cannot be efFeded by thofe 

 transformations. 



But the equation, x' + px' + qx + r=: by transforming the 

 roots into their reciprocals, and freeing the firft term from a coeffi- 

 cient becomes, x' + qx'' +prx + r'' ^0 therefore if in the propofed 

 equation q' = 3/^, then by transforming the roots into their re- 

 cio'-ocals, and freeing the firft term from a co-eifiwient the equation 

 will be reduced to the required form. 



Any cubic equation being propofed, there is a quantity, by 

 which if the roots are encreafed (or diminifhed) q' will become 



equal 





