ALGEBRA 



Then if X is large enough so that c p x is large compared to Cp+i^, Ci x , C2 X , 

 c p x approximately satisfy the equation: 



R Q uP - RiuP- 1 + R 2 uP~ 2 - . . . . R p = o 

 and Cp+i x , Cp+z*, > C * X approximately satisfy the equation: 



Therefore when X is large enough the given equation breaks down into a number 

 of simpler equations. This stage is shown in the process of deriving the suc- 

 cessive equations when certain of the coefficients are obtained from those of 

 the preceding equation simply by squaring. 



REFERENCES: Encyklopadie der Math. Wiss. I, i, 3a (Runge). 

 KBAIRSTOW: Applied Aerodynamics, pp. 553-560; the solution of a numerical 

 equation of the 8th degree is given by Graeffe's Method. 



1.270 Quadratic Equations. 



X 2 + 2dX + b = O. 



The roots are: 



If a 2 > b roots are real, 



a 2 < b roots are complex, 

 a 2 = b roots are equal. 



1.271 Cubic equations. 



(1) x 3 + ax 2 + bx + c = o. 

 Substitute 



(2) x = y - 



O 



where 



---b 

 3 



ab 2 , 



20 = a? c. 



3 27 



Roots of (3) : 



If p > o, q > o, # 2 > p* 



cosh = 



