689] 
BY A REAL CORRESPONDENCE OF TWO PLANES. 
323 
41 
•2 
to its original position, upon the corresponding point P a returning to its original 
position and sheet. This is, in fact, Riemann’s theory, only instead of the points P' 
we must speak of the values x' + iy' of the irrational function of x + iy. 
18. Everything is of course symmetrical as regards the two planes; we have 
therefore, in the second plane, a system of points V and of barriers, and in the first 
plane a system of points (IT) and of counter-barriers. To a given point P' in the 
second plane there correspond m points P in the first plane; and we can (the 
determination depending on the system of barriers in the second plane) assign to the 
in points suffixes, thereby distinguishing them as the corresponding points P 1} P 2 ,..,P m . 
And we may imagine the second plane as consisting of m superimposed planes or 
sheets, say the sheets 1, 2, 3,.., m; the general theorem then is that to a point P 
or P' in either plane, considered as a point P a or P a ' in the sheet a. or a', there 
corresponds in the other plane one and only one point P a ' or P a >; and that the first- 
mentioned point in either plane moving continuously in any manner (subject to the 
proper change of sheet), the corresponding point in the other plane will also move 
continuously, and will return to its original position and sheet, upon the first- 
mentioned point returning to its original position and sheet. 
19. In all that precedes it has been assumed that, to a branch-point V, there 
correspond two united points represented by (W') and n — 2 distinct points W'; the 
cases of a point {W') composed of three or more united points, or of the points W 
uniting themselves in sets in any other manner, would give rise to further specialities.