SEMINAR NOTES 399

approaching one and ascertain the effect of a passage round, etc. Just what is meant by this will be made clearer after the mathematical exposition.

Consider the function defined by the cubic equation

K/3 — W -\- Z = . . . .

"For each value of z, w has in general the three values w,, w,, w^.

2

But the last two of these become equal when z^-- . — . At this point

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we have w,^■w^=^^' ^. If now we assume that the variable 2 changes

continuously, or that the point representing it describes a line, then

each of the three quantities w, , w,, w^ likewise changes continuously,

or the three corresponding points describe three separate paths. But

2 when 2 passes through the point 2= — , both functions w , and w,

assume the value \ \, hence the two lines w, and w^ meet in the point

1/^. At the passage through this, therefore, w, can go over into w^,

and w^ into w,, without interruption of continuity ; indeed, it remains

entirely arbitrary on which of the two lines each of the quantities

w, or »3 shall continue its course. In this place a branching of the

lines described by the quantities w, and zf, takes place, hence Riemann

has called those points in the 2-plane at which one value of the

function can change into another, branch-points.'

"In Fig. A the three w,, w,, w^ are drawn for the case when z

describes a straight line parallel to the j-axis and passing through the

2 branch-point e^ — 7=. The az-points which correspond to the 1 27

2-points are denoted by the same letters with attached subscripts i, 2,

3. Let us follow the path of only one of the quantities, say Wy This

I describes the line b^ c^ d^ and approaches the point ^3 = ^,= — — , as z

2 approaches the point e^^ . — along the line bed. Should 2 now

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nass through the point, w^ could continue its course from e^^e^^^v \

on either of the two paths ^3/3 ^3 h^ or e,f,g, h„ on which one as well

'Definition — "A point at which either a discontinuity occurs or several function values become equal is called a branch-point when, and only when, the function changes its value in describing a closed line around this and no other similar point."

For illustrations of the test applied, see Durfege, p. 42.