THEORY OP EQUATIONS. 



895. If a, /2, y, 8 be the roots of a biquadratic, and the 

 equation be solved by putting it in the form 



the values of 2b are 



Py + aS 

 those of 4c are 



( / S + y _ a _8), (y + a-0-S)', (a + /3-y-S) 2 ; 

 and of (I are 



ffy-aS ya-/?8 a/3-y8 



)3 + y-a-6' y + a-/3-8' a +/J-y-8* 



89 G. In the method of solving a biquadratic in x by sub- 

 stituting x = my + n, and making the resulting equation in y 

 reciprocal ; prove that the three values of n are 



Py-oS ya-pS a/?-y8 



/3 + y-a-8' y+a-0-8' a +/J-y-8' 



and those of m* are 



-.----.8) 



a i A 7> S being the roots of the biquadratic. 



897. Prove that the equation 



3 4 + 8x*- 6a^- 24a: -f r= 



\vill have four real roots ifr<-8>-13; two real roots if 

 r > - 8 < 19 ; and no real roots if r > 19. 



898. Prove that the equation 



will have two equal roots if 



899. If /(*)= (*-<,)(*-,) ... (x-a m ) ; and a,, a., ... a. 



be all unequal; thru will 



a,"*-' /*'-' 

 /'<,) /W* 



