If one assumes that the average flux did not change between measurements, a mass-distribution curve is obtained which relates the flux of particles larger than a given radius to the inverse 7/2 power of the radius.

Consider a simple, closed, plane curve C which is a real-analytic image of the unit circle, and which is given by Af.

In the following paper it is shown that in a certain definite sense, exactly an odd number of squares can be inscribed in every such curve which does not contain an infinite number of inscribed squares.

If the vertex is at Af, and if the interior of C is on the left as one moves in the direction of increasing t, then every such corner can be found from the curve obtained by rotating C clockwise through 90-degrees about the vertex.

But Af is just the curve Af translated without rotation through a small arc, for Af is always obtained by rotating C through exactly 90-degrees.

The arc is itself a segment of an analytic curve.

Then in 2 we show that any line involution with the properties that ( A ) It has no complex of invariant lines, and ( B ) Its singular lines form a complex consisting exclusively of the lines which meet a twisted curve, is necessarily of the type discussed in 1.

The order of this congruence is Af, since Af secants of a curve of symbol ( B ) on a quadric surface pass through an arbitrary point.

For the lines of any plane, **yp, meeting Q in a conic C, are transformed into the congruence of secants of the curve C' into which C is transformed in the point involution on Q.

Each generator, **yl, of Af is also exceptional, for each is transformed into the entire congruence of secants of the curve into which that generator is transformed by the point involution on Q.

This curve is of symbol Af since it meets **yl, and hence every line of Af in the Af invariant points on **yl and since it obviously meets every line of Af in a single point.

Hence C' is a Af curve on Q.

By ( 1 ), the image of this pencil is a ruled surface of order Af which is met by the plane of the pencil in a curve, C, of order Af.

To avoid this contradiction it is necessary that C be composite, with the secant of **zg and a curve of order Af as components.

We now observe that the case in which **zg is a Af curve on a quadric is impossible if the complex of singular lines consists exclusively of the lines which meet Aj.

This contradicts the preceding observations, and so, under the assumption of this paper, we must reject the possibility that **zg is a Af curve on a quadric surface.

Continuing with the case in which **zg is a Af curve on a quadric Q, we first observe that the second regulus of Q consists precisely of the lines which join the two free intersections of **zg and the planes through any one of the multiple secants.

We assume that average total unit cost in the relevant region of operation is constant with respect to quantity produced ( the average cost curve is horizontal, and therefore is identical with the marginal cost curve ), and is the same for every firm ( and therefore for the industry ).

Bridgeman's piece of water called The Serpentine, formed by damming the little Westbourne that flowed through the park was not truly in the Serpentine " line of beauty " that William Hogarth described, but merely irregular on a modest curve.

However, while Keynes's formulation of a liquidity trap refers to the existence of a horizontal demand curve for money at some positive level of interest rates, the liquidity trap invoked in the 1990s referred merely to the presence of zero interest rates ( ZIRP ), the assertion being that since interest rates could not fall below zero, monetary policy would prove impotent in those conditions, just as it was asserted to be in a proper exposition of a liquidity trap.

In the IS-LM-BoP graph, the IS curve has been shifted exogenously by the fiscal authority, and the IS and BoP curves determine the final resting place of the system ; the LM curve merely passively reacts.

When animators or interface designers talk about the " physics " or " feel " of an operation, they may be referring to the particular Bézier curve used to control the velocity over time of the move in question.

As the curve is completely contained in the convex hull of its control points, the points can be graphically displayed and used to manipulate the curve intuitively.

Affine transformations such as translation, and rotation can be applied on the curve by applying the respective transform on the control points of the curve.

A common adaptive method is recursive subdivision, in which a curve's control points are checked to see if the curve approximates a line segment to within a small tolerance.

A Bézier curve is defined by a set of control points P < sub > 0 </ sub > through P < sub > n </ sub >, where n is called its order ( n

The points P < sub > i </ sub > are called control points for the Bézier curve.

* The curve is a straight line if and only if all the control points are collinear.

* Some curves that seem simple, such as the circle, cannot be described exactly by a Bézier or piecewise Bézier curve ; though a four-piece cubic Bézier curve can approximate a circle ( see Bézier spline ), with a maximum radial error of less than one part in a thousand, when each inner control point ( or offline point ) is the distance horizontally or vertically from an outer control point on a unit circle.

More generally, an n-piece cubic Bézier curve can approximate a circle, when each inner control point is the distance from an outer control point on a unit circle, where t is 360 / n degrees, and n > 2.

* Every quadratic Bézier curve is also a cubic Bézier curve, and more generally, every degree n Bézier curve is also a degree m curve for any m > n. In detail, a degree n curve with control points P < sub > 0 </ sub >, …, P < sub > n </ sub > is equivalent ( including the parametrization ) to the degree n + 1 curve with control points P '< sub > 0 </ sub >, …, P '< sub > n + 1 </ sub >, where.

He had style: he held his reins in a loose bunch at the third button of his checked Epsom surtout, and when the horses leaned at a curve, as if bent by the force of a gale, he leaned with them.

The upper curve denotes the mean plus one standard deviation, and the lower curve represents the mean minus one standard deviation.

Thus, a child's Skeletal Age `` dots '' may be classified as `` advanced '' when they appear above the middle curve, `` moderate '' when they appear immediately above or below the middle curve, and `` delayed '' when they appear below the lower curve.

The Skeletal Age curve in the lower portion of Figure 2 shows that slowing may have occurred for her during the prepubescent period.

A square inscribed in a curve C means a square with its four corner points on the curve, though it may not lie entirely in the interior of C.

For any such square the middle corner of these will be called the vertex of the square and the corner not on the curve will be called the diagonal point of the square.

The set of intersections of Af, the rotated curve, with the original curve C consists of just the set of forward corner points on C corresponding to the vertex at Af, plus the vertex itself.

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

These are defined by a simple involutorial transformation of the points in which a general line meets a nonsingular quadric surface bearing a curve of symbol Af.

Let Q be a nonsingular quadric surface bearing reguli Af and Af, and let **zg be a Af curve of order K on Q.