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A tangent point Q in the C-plane occurs when C and Af are tangent to one another.

A continuous change in T through an amount E results in a translation along an analytic arc of the curve Af.

There are three possibilities: ( A ) Af remains tangent to C as it is translated ; ;

( B ) Af moves away from C and does not intersect it at all for Af ; ;

( C ) Af cuts across C and there are two ordinary intersections for every T in Af.

The first possibility results in a closed interval of tangent points in the f-plane, the end points of which fall into category ( B ) or ( C ).

In the second category the function Af has no values defined in a neighborhood Af.

In the third category the function is double-valued in this interval.

The same remarks apply to an interval on the other side of Af.

Again, the analyticity of the two curves guarantees that such intervals exist.

In the neighborhood of an end point of an interval of tangent points in the f-plane the function is two-valued or no-valued on one side, and is a single-valued function consisting entirely of tangent points on the other side.