ON THE ROOT OF A MONOGENIC FUNCTION 339 



m ^ l/(o) I at this point, this is a curve which has a multiple node at the point 

 with n distinct loops inside C, in each one of these loops / {z) has a root. 



6. In case at the critical point .r, y both f{z) and f{z) are zero, then the 

 first derivative of f which does not vanish is the fourth, which at the point 

 is given bj^ 



*^ dr' \dr-/ '^ \dry ' 



which is positive when f"(z) =t= 0, and at the point f is a minimum. 



In general if at x, y we have f'(z) = when r = 0, 1, . . ., ?i — 1 

 then the first derivative of f which is not zero there is given by 



1 d-"f _ ( d''u\- fd^v 



n (n + 1 ) dr'-" \ dr" / \ dr" 



which, being positive when f^iz) 4^ 0, there results a minimum for f at 



X, y- 



7. If a function f{z)- is regular at all points of a region S bounded by a 

 simple closed curve s, and if at any point P inside s the modulus of f{z) is 

 less than the least value of the modulus on .s then f{z) has at least one root 

 iaside s. For if m is the least value of \f(z) I along s then 



\Kz)\ = m 



is a closed contour in the region S and contains P. 



8. The functions u and v being conjugate 



c)r-" d^/-"' dy"" d.r'" 



The results of § 4 show that 



- — = -:— — cos n5 + ( — 1 ) ^T- sm nd. 

 dr-" ox-" oy-" 



This changes sign as 7i6 changes, and therefore, as is well known, the har- 

 monic function cannot have a maximum or minimum value at any point in 

 the X, y plane. 



15. On the Root of a Monogenic Function inside a closed Contour along which the 

 Modulus is Constant. By W. H. Echols. Vol. I, pp. 335-339. Price, $0.25. 



