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suppose that at the point under discussion, not only = 0,

and (^) = 0, but also \dy'

(£)-* <147)

then the value of -3? again assumes the form Q, and the numerator and denominator of it must be differentiated; in which operation however it is to be borne in mind, see Art. 139, that

T£ does not vary with x and y near to the point, and is therefore

to be considered constant; the true meaning and effect of these successive differentiations being as follows. Several branches of the curve have certain consecutive points in common, and certain elements in common; whilst therefore we are considering the curve, as to its continuation at one or more of these common points, it is indeterminate to which branch of the curve the points and elements belong, and therefore we must pass on from these common points to those contiguous ones which are on different branches; in which case the tangent lines drawn through these become separate for each branch, and the direction of each thereby becomes determined. Let the reader try to draw for himself an infinitely magnified diagram of such points and curves in the same manner as we have drawn fig. 56.

Differentiating therefore the numerator and denominator of the right-hand member of equation (125), and dividing through by dx.

,dsF\ g / d3T \ dy 1 d3¥ \ dy"

dy ^dx3' ~\dx'2dy'dx ^drdy2' dx2

-j— = — 1 ;(148)

dx id3r\ dy2 1 d3p \dy 1 d3v \

{dy3* dx2 + \dxdy2) dx + ^dx^dyl whence, multiplying and reducing,

(^)^+3(rf^)d^+3(^^)£+(^)=0; (149)

which is a cubic equation in —, and therefore has three roots;

shewing that three branches of the curve pass through the point, which is accordingly called a triple point; the three branches at which the curve meets the coordinate axes, by finding the values of x which render y = 0, and the values of y which render x = 0; and let the change or continuation of sign be examined in order to determine whether the curve passes from below to above the axis of x, or vice versd; and whether it passes from the left to the right of the axis of y, or vice versd; and whether it touches the axes; and if it cuts the axis, let the value of

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being all in the plane of reference, or one in and two out of the plane, according as the roots of (149) are all real, or one real and two impossible.

As the criteria of this division lead to a long and complicated expression, it is needless to investigate it; and, moreover, as the

determination of the several values of ~ , corresponding to the


several branches of the curve, is not difficult, we shall only add an example.

Ex. 1. To determine the nature of the point at the origin of the curve whose equation is x*—ay.r2-^ by3 = 0.

dy 4x3—2axy n ... dx = ax*-Uf = 0' at the °n^n'

_ (I2x2 2ay)dx 2axdy _ Q

2axdx—6bydy ~ 0''

24xdxi—4adydx 4adydx

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■ vra "T I i

dx dx

and therefore at the origin there is a triple point, as three branches of the curve pass through it, of which one touches the axis of x, and the other two are inclined to it at angles

whose tangents are + (~) ; see fig. 69.

256.] Similarly if all the third partial differential coefficients vanish at the point under discussion, we must differentiate again the numerator and denominator of the right-hand member of (148); by which means we shall obtain a biquadratic expresdv

sion in indicating that four branches of the curve pass


through the point, which is therefore called a quadruple point.

257.] Such is the general theory of multiple points; of which the analytical character is the vanishing at the point of successive partial differential coefficients of the implicit equation to

the curve. That such must vanish, if many branches pass through the same point, may thus be shewn a priori.

Let a curve be such that, when x = a + A, y has many values, or, to borrow language from the theory of equations, the equation formed in powers of y has many unequal roots; but when x = a several of these values of y become equal, say y = b; then, in this case, as many roots as become equal which before were unequal, so many branches of the curve pass through the point; and thus in the equation many equal factors will be multiplied together, which produce a factor of the form (y b)". By similar reasoning we may prove that at such a point many factors involving x, which at other points are unequal, become equal; and we have a factor of the form (x—a)m, m and n being some numerical quantities greater than 1. Now since differentiation diminishes the exponent of such a quantity only by

unity, it is plain that will, at the point in question, have a factor of the form (x—o)m_1, and therefore will = 0. Similarly {^) wdl have a factor of the form (y b)"-1, and will

= 0 also; and according to the numerical magnitudes of m and n, will be the number of branches passing through the point, and the number of successive partial differential coefficients which = 0, for the values x = a, y = b.

Section 5.—On tracing curves by means of their equations.

258.] As this treatise is intended in a great measure for didactic purposes, I shall insert at some length an account of the most simple processes by which we can delineate a curve expressed by a given equation: and herein we shall introduce simultaneously and combine the methods which have been in the preceding sections separately investigated for the discovery of singular points of curves. The object of this inquiry is twofold: to give expertness, firstly, in the analysis of an equation and of its derived functions; and, secondly, in the translation of these circumstances into their corresponding geometrical forms. AU curves however we cannot trace, any more than find the equations to all figures of a character however complicated; the problem is as general as the solution of all equations; and

Price, Vol. i. 3 E

therefore what follows is to be taken as an explanation, and as a specimen, of the means we possess of discussing some few simple cases which are for the most part algebraical.

1) If the equation admits of being simplified by a change of origin, or by turning the axes through any angle, or by transforming the equation into its equivalent in terms of polar coordinates, let such a change be effected before we begin the analysis. Thus, for instance, the equation .r22ax + y2 + 2by = 0, admits of being discussed more easily when for x we write x + a, and for y, y b: whereby the result becomes x^ + y* = a2 + b2. Similarly the curve whose equation is (aP + y2)^

= a tan-1


is more easily traced when it is put in its equivalent polar form, r = ad. The means of tracing polar curves will be discussed in the next chapter.

2) If the equation to a curve admits of being put in the form /., . , , , ,

y = /(*) ± 0 (x),

in which case, as before observed, y =f(x) is the equation to a curve diametral to the curve to be- traced, it is most convenient first to trace the diametral curve, and then to increase and diminish its ordinates by the quantity <f> (x) corresponding to the several values of the abscissa to the curve y =f(x).

Thus, for instance, in the discussion of the general equation of the second degree,

ay2 + bxy + cx2 + dy + ex +f = 0, (150) let the equation be solved for y; whence we have

= - £-x- ± {(b2-4ac)x2 + 2(bd-2ae)x + d*-4,af}i; (151)

V = ~ 2~aX ~ 2~a' 1S *'16 e1uat'on to a straight line, and therefore the ordinate to the curve is the ordinate to a straight line increased and diminished by equal quantities; the most convenient method therefore of tracing the curve is first to construct the straight line, and then to add to and subtract from its ordinate such a quantity as arises from an examination of the latter part of (151).

3) Let the equation to the curve, if possible, be put in the explicit form y = f(x); and let all the points be determined

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^ be examined at the point of section, so as to determine the

angle at which it cuts. And if, for all values of x from + x to — oc , y is unaffected with + «J , the curve extends infinitely in both directions in the plane of the paper; but if at any point, say x = a, y = b, the equation is such as on one side of that point to be affected with +, and on the other side with + v —, then at that point the curve leaves the plane of the paper. Suppose that at such a point there is only one branch of the curve, so that the symbol of " impossibility" does not arise from the extraction of the square root of a negative number, then there is what is by French writers termed a point d'arret; or, as we may conveniently call it, a point of abrupt termination; and the branch has only one tangent. Such however can only arise from a discontinuous function, or from such functions as those for which Maclaurin's Theorem fails. Thus, if the equation is y = x2 log x, y = 0, when x 0;

by virtue of Ex. 2, Art. 126: also -r- = 0, when x = 0; hence


the curve passes though the origin, and touches the axis of x, aud is in the plane of the paper on the positive side of the axis of y; but as the logarithms of negative numbers are, see Art. 67, affected with + \f . the curve is out of the plane of reference on the negative side of the same axis; and therefore there is at the origin a point of abrupt termination. The above curve is traced in fig. 70 as far as it exists in the plane of the paper;

AO = 1.

If however at the point where the ordinate becomes affected with +-</—, two branches pass into another plane, there is either a cusp or a salient point, according as the two branches have the same or different tangents. The distinctive characters of these points depend on the corresponding value or

values of -jr. And if the equation to the curve is satisfied

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