Manchester Memoirs^ Vol. xliii. (1899), No. 7- 



VII. A New Version of Argand's Proof that every 

 Algebraic Equation has a Root. 



By Prof. Horace Lamb, M.A., F.R.S. 



Received and read March jth, iSgg. 



I. The classical proofs of the theorem in question are 

 for the most part long, and to some minds not very 

 attractive. It may be worth while to indicate how by 

 means of a theorem which, although of a ' transcendental' 

 character, is (in another connection) thoroughly familiar 

 to most mathematicians, the matter can be presented in a 

 very simple form. 



Denoting by f{z) a rational integral function of z let 

 us write, as usual, 



z = x->riy = re . . . . ( i ), 

 f{z) = u^iv = Ri® . . . (2), 



so that R and 9 denote the 'modulus' and 'amplitude', 

 respectively, of/(^), viz., 



R^ ^{u^^v"), e = tan-i- . . (3). 



Since R becomes infinite with r, and cannot be negative, 

 there must be some finite point in the plane xy at 

 which R attains a lower limit. This limit cannot be 

 other than o, for then log R would have a finite minimum 

 value. This is (by a well-known corollary to Green's 

 theorem) impossible, since the function ^ = log R satisfies 

 the equation 



^V'^° .... (4), 



and is (together with its first and second derivatives) finite, 



June 6th, i8gg. 



