1890.] to the Theory of Functions. 127 



therefore at every point of this surface u and v will each have a 

 single definite value, the only exceptions being the points 

 corresponding to infinite values of w, at which (since w is an 

 algebraical function) u and v will each take all possible values. 

 If then on this surface the curves u = c(— oc < c < oo) are drawn 

 the separate curves will nowhere meet each other except at the 

 points where w is infinite, and through these points all the curves 

 of the family will pass. The same remark applies to the w-curves. 

 The Niemann's surface on which the curves have been drawn 

 consists in its simplest form of n superposed infinite planes, the 

 continuity between different planes being provided for by cross- 

 cuts (to use Clifford's translation of " Verzweigungsschnitte ") con- 

 necting properly the branch -points. If now these n planes be 

 regarded as transparent so that the curves may be seen as though 

 lying in one plane, the result will be the same as though they 

 had been so drawn originally. It is then at once clear that 

 through every point of the single x, y plane n it-curves and n 

 v-curves will pass. [It is only by properly taking the curves in 

 pairs so that u + iv is a root of f{z, w) = for the value of x + iy 

 considered that the u- and ^-curves will cut at right angles.] 

 The only exception will be that through the points x + iy, which 

 make one or more values of w infinite, all the curves of both 

 systems will pass. The branch-points, i.e. the points x + iy which 

 make the equation f= have equal roots, will be distinguished 

 in this way; that whereas generally the values u v u 2 ...u n of the 

 parameters of the n w-curves passing through a particular point 

 are all different ; if the point is a branch-point at which r roots 

 of the equation f= become equal, r branches of one curve u t 

 and n — r other curves u r+1 ,...u n will pass through the point; as 

 well, of course, as r branches of a curve v 1 and r other curves 

 v _,_,... v . It is obvious at once that such a family of curves can- 

 not have what is usually called an envelope (real), for this would 

 clearly necessitate the Riemann's surface consisting of an infinite 

 number of sheets. 



The point in which Mr Brill seems to have gone astray is the 

 following. In § 3 of his paper he says that if the curves v and 

 v + dv intersect, the necessary condition is 



/ { u + iv] =f [u + i (v + dv)} ( A) ; 



[the equation between z and w in the paper referred to is written 



This clearly is not necessary, for the curves will intersect if 



f{u + iv}=f{u' + i(v + dv)} (B), 



where u and ii are different. 



11—2 



