4 MATHEMATICAL FORMULAE AND ELLIPTIC FUNCTIONS 



1.221 It follows from 1.220 that f(ti) = R. This gives a convenient way of 

 evaluating f(x) for x = h. 



1.222 To express f(x) in the form : 



/(*) = A Q (x - h} n + Ai(x - h) n ~ l + ....+ An-i(x - h) + A n . 



By 1.220 form f(ti) = A n . Repeat this process with each quotient, and the 

 last term of each line of sums will be a succeeding value of the series of co- 

 efficients A n , An-i, , AQ. 



Example : 



/(*) = 3* 5 + 2x* - Sx z + 2x - 4 h = 2 



32 = Al 



3 = A * 

 Thus: 



24X + 50 

 K -/(*)- # 



*) = 3(* ~ 2 ) 5 + 32(* - 2) 4 + 136(0: - 2) 3 + 280^ - 2) 2 + 274(0; - 2) H- 96 



TRANSFORMATION OF EQUATIONS 



1.230 To transform the equation f(x) = o into one whose roots all have their 

 signs changed: Substitute -x for x. 



1.231 To transform the equation f(x) = o into one whose roots are all multi- 

 plied by a constant, m : Substitute x/m for x. 



1.232 To transform the equation f(x) = o into one whose roots are the 

 reciprocals of the roots of the given equation : Substitute i/x for x and multiply 

 by x n . 



1.233 To transform the equation f(x) = o into one whose roots are all increased 

 or diminished by a constant, h : Substitute x h for x in the given equation, 



