338 UNIVERSITY OF VIRGINIA PUBLICATIONS 



-^ — = t ^- + m :— u = 0, -^ = [I ^ + m—-] V = 0. 

 dr" \ bx by dr" \ bx by 



Also in general 



(V^ rv / d^u , „ du d^~^u , , du d''~^u 



dr" \ dr' dr dr'~^ dr dr''~^ 



( d''v „ dv d"-h dv d'-h 



^ V d^ ^ "''Jrlti^^ ' ■ ■ + 5^ *^ 



These derivatives vanish at .r, ?/ for p = 1, . . ., n — I, while the nth 

 derivative at that point has the value 



, d"f d^u , d"v 



dr^ dr^ dr" 



I b"u , b^v \ „ , / b^u b^v \ . 



= I u — — + V — — I cos 7id + v u -=- — I sm nd, 



\ bx" bx" ! \ ox" O.T" / 



= + ( M ^^ — + 1}^ — I, when tiQ = 

 \ o.r" o.r" / 



/ £)"2t I c)"y \ , . 



— — \u - — + V - — I , when m = t 

 \ bx" bx" I 



b"u . „ , _ S"y _ 



= V — — sm nd, when u = 0, ^ — = 0, 

 bx" bx" 



and the last value changes sign with iid. Hence it is impossible for x, y 

 to be a pomt at which f has its minimum. Therefore f(z) must have 

 a root at a;, y unless f(z) is merely a constant throughout the region C. 



Cor. A polynomial /(s) of degree n. cannot be zero for any value \z\ > 7ip 

 where p is the absolute value of the quotient of the coefficient having the 

 greatest absolute value to that of z". Hence for an assigned a>np 



1/(^)1 = !/(«)! 



is a simple closed contour in which /(s) has at least one root, and therefore 

 as easily follows n roots. 



5. If at a second point in C the first n — 1 derivatives of /(a) vanish and 

 there f(z) =t= 0, then the nth derivative of f{z) vanishes there for n distinct 

 values of 6 and if 



f = m^ = M^ + v,^ 



