276 



BELL SYSTEM TECHNICAL JOURNAL 



THE ISOGRAPH 



The Isograph was developed in Bell Tele- 

 phone Laboratories to find mechanically 

 the complex roots of polynomials of high 



degree. Let the polynomial to be factored 



n n n 



be p{z) = Sfflj-z' or 2o,t^ cosjQ + iZaf^ sinjG 



i 



if 3 = r (cos9 + isinG). The isograph 

 maps the complex values of p{z) as the 

 variable describes the circle |z| = r. This 

 graph loops the origin once for each root 

 smaller in absolute value than r. The 

 number of roots between trial values of r 

 is determined by counting loops, and by 

 interpolation a value of r is found for 

 which the graph passes through the origin. 

 This value of r and the corresponding value 

 of 9 define the real and imaginary parts of 

 a root. 



