QUADRATIC EQUATI0>;S, 367 



Again, :c'"f^^ xx+yzzs^ — 2/»XJ by multiplication. 



Or X ' -\->:y X .v -j-y -\-y^=:s'^ — 2 j/, 

 Or x^-{'Sj>-\-y^=:s^ — 2sj> by substituting s/> for its equal 



xyXx-^y ; 

 And therefore a?'-|-j>'=:j' — 3jr/r? sum of the cubes. 



In like manner, .■v'+j* x^+j— j' — 3Jf/'XJ'by multiplication. 



Or x'^-\-xyXx''+y'' +y*=s'^—3s^p. 

 Or iv*4-/x J^ — zjf'^-y'^zzs'*- — p^p by substituting / x J^— 2^ 

 for its equal xyX^^-\-y^ ; 



And consequently, x"^ +y'^—s'^'- ^s'p—pxs^-^zp z^s"^ — 4^* 

 /4-2/*^=: sum of the biquadrate.s, or fourth powers ; and 

 so on, for any power whatever. 



10. The sum (a) and the sum of the squares (l^) of four 



numbers in geometrical progression being given j to find 



those numbers. 



Let X and y denote the two means, 



a; ' v^ 



Then will — and — be the two extremes, by the nature 



y PC 



of proportion. 

 Also, let the sum of the two means zn/, and their 



product =: p. 

 And then will the sum of the two extremes :=za — j- by 



the question. 

 And their product rz /> by the nature of proportion. 

 Cx- ^y^ -s^-2p -1 



Hence < a;-* ^* ___ :* > by the last problem, 



Cx- ^y^ -s^-2p -1 



And x^-fj;*-j r + -^ — J* + ^— j| —^pzzh by the quest- 

 ion, y ^' 



Again, 1- -^ zza — s by the question, 



y X 



Or 



