128 Mr Burnside, On the Theory of Functions. [Nov. 24, 1890. 



In the light of the above geometrical reasoning the condition 

 (A) is, except at a branch-point, equivalent to supposing that the 

 imaginary part of w has the two different values v and v + dv at 

 one point of the Riemann's surface, which is inconsistent with w 

 being a single-valued function on the surface (unless the point is 

 an infinity of w). 



From the condition (A) Mr Brill at once deduces the relation 

 f'(ti + iv) = 0, 



on which the statements mentioned at the beginning of this note 

 are based. 



The condition (B) implies that the imaginary part of w has the 

 two values v and v + dv at points corresponding to the same x + iy 

 on two different sheets, say 1 and 2, of the Riemann's surface. The 

 values of v on each sheet are continuous and hence on sheet 2 

 there must be a branch of the curve v indefinitely near the branch 

 v + dv. This branch v on sheet 2 and the branch v above con- 

 sidered on sheet 1 will intersect, generally at a finite angle, when 

 the curves are drawn on a single plane ; and hence the locus of 

 real intersections of consecutive curves, which it was shewn above 

 could not be an ordinary envelope, is a locus of double points. 

 It is to be noticed that corresponding to the double point con- 

 sidered on the v-curve there will generally not be one on a it-curve. 



Mr Brill applies his equation 



f (u + iv) = 



to discuss not only the real but the imaginary intersections of the 

 curves in question. To this purpose the equation appears to me 

 to be entirely inapplicable. 



In considering the imaginary intersections of two ^-curves, 

 the problem in hand becomes one of functions of two complex 

 variables defined by equations like 



f t 0, V, u) = 0, 

 where x and y may both be complex. 



If the attempt be made to deal directly with imaginary values 

 of x and y, say x t + ix 2 , y t + iy 2 in the original equation defining 

 the function, £ becomes x x — y 2 + i (^ 2 + 2/i), and in the analytical 

 work all trace of the original point is lost, as it is impossible to 

 pass back from the two real quantities x x — y 2 and x 2 + y x to the 

 original complex quantities which gave rise to them. 



