Points d'Arrêt or Points de Rupture.

122. An algebraic curve never stops abruptly in its course, that is, it never possesses singular points of the kind called by French writers points d'arrêt or de rupture. Such points are however of frequent occurrence in transcendental curves. instance, in the curve belonging to the equation

For

The impossibility of the existence of such points in curves represented by algebraical equations depends upon the fact that impossible roots enter algebraical equations, involving one unknown letter, by pairs. Suppose in fact that, when x is equal to ah, y has an impossible value for each small value of h however small, and that y has a possible value when x is equal to a + h. Then, when x passes from a h to a +h, one value of y, and therefore, by the nature of algebraical equations, two values of y, the two values being of the forms a ± √(B), must change into possible ones, which will evidently, in consequence of the correspondency of their values, be equal to each other when h is indefinitely diminished. The existence of two equal values of y, corresponding to the value a of x, shews that there is no abrupt termination of the curve at the point of which the abscissa is a.

Points Saillants.

123. A point saillant is a point of a curve where two branches of the curve stop abruptly and have tangents inclined to each

other at a finite angle. Such points are frequently to be met with in transcendental curves, but can have no existence in curves corresponding to algebraical equations.

Ex. 1. Take the curve of which the equation is

The origin of coordinates is therefore a point sailiant: the branch corresponding to positive values of the abscissæ touches the axis of x, while the tangent to the other branch at the same point is inclined at an angle of 45° to this axis.

There will be a point saillant at the origin of coordinates: the directions of the two branches at this point being defined

We have observed that an algebraical curve does not admit of points saillants. This may readily be shewn. Suppose, in fact, that between the equation to the curve

we eliminate y: we may then obtain an algebraical equation between x and y', free from radicals and fractional forms. If then we conceive a curve to be constructed, of which the abscissa shall be always equal to x and the ordinate to y', this curve can have no point d'arrêt, which would necessarily be the case if the primitive curve had any point saillant.

Branches Pointillées.

124. We occasionally meet with equations, the geometrical loci of which consist, either entirely or in part, of a series of conjugate points, forming branches pointillées.

It is evident that y will be always impossible when sin

()

has any negative value or any positive value less than unity; and therefore, unity being the greatest value of the sine of an angle, we must have

λ being an integer. Thus we see that the geometrical locus of the equation consists of a series of conjugate points arranged along the axis of x at a common interval equal to 2πa, the axis of abscissæ being thus a branche pointillée.

CHAPTER IV.

CONCAVITY AND CONVEXITY OF CURVES AND POINTS OF INFLECTION.

Conditions for Concavity and Convexity.

125. THE object of this chapter is to investigate the condition that a curve, of which the equation is given, may at any assigned point turn its concavity or convexity towards either of the coordinate axes, and to determine those peculiar points of the curve, called points of inflection or of contrary flexure, at which concavity is succeeded by convexity, or conversely.

In fig. (1) the curve AB is concave at P towards the axis of x, and convex towards that of y; in fig. (2) there is a point of inflection at P, the curve being concave towards the axis of x at each point in the arc AP, and convex at each point of the arc PB.

Let,', be the inclinations of the tangents of the curve at P, P', in fig. (1) to the axis of x, P' being a point near to P: then, as we pass through P from A towards B, it is evident that tan✈ will keep continuously decreasing: hence, x, x', being the coordinates of P, P',

and this will be true however near P' may be to P; and therefore, proceeding to the limit, we see that, as the condition of

If the curve were convex at P towards the axis of x, it is evident from like reasoning that

dzy

be negative,

dx

the axis of x at a point x, y, it is necessary that

day dx2

and, that it may be convex, that be positive.

It is easily seen that the conditions which we have shewn to be necessary are also sufficient conditions for concavity and convexity. For it is evident that the curve will be concave or convex towards the axis of x at a point (x, y) accordingly as

is negative or positive for all indefinitely small values of y'

nearly equal to y, that is, proceeding to the limit, as negative or positive.

dzy

is dx2

If y be negative, or the point P be on the opposite side of the axis of x, then, as may be seen simply by changing the sign of y throughout, we must have, for concavity,

Without regarding the sign of y, we may state generally that the sufficient and necessary condition for concavity is that