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and for a value a h impossible, or vice versa, the curve stops short at the point in question, and is doubled back on itself, forming what is called a cusp. The cusp is said to be of the first species or a ceratoid* when the branches touch the common tangent on opposite sides; and of the second species or a ramphoid † when they touch on the same side. These may be distinguished by the consideration that in the first the values of " ? are of opposite, and in the second of the same signs. It is to be observed that at a cusp the two branches of the curve never make with each other an angle the trigonometrical tangent of which is of finite magnitde: we cannot properly say that the angle itself is infinitely small, as in fact it is equal to two right angles, the inclination of the one branch of the curve being measured in a direction opposite to that of the other. Although the condition of qy when of the form having impossible values always indicates a conjugate point, yet it may happen that and any number of the differential coefficients are possible at a conjugate point. In such cases the impossible branch of the curve does not pierce the plane of the axes, but touches it at the conjugate point, the order of contact being that of the highest differential coefficient which is possible. To determine with certainty whether a point be or be not a conjugate point or a cusp, it is always necessary to try whether the equation to the curve gives possible values for both variables on each side of the point in question.

If some of the values of be possible and some impossible for the given value of x, there is a conjugate point situate on the curve ; that is, a branch in the impossible plane pierces the plane of reference in a point through which there passes a possible branch of the curve.

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* Képas, a horn. † 'Puupos, a beak.

For a fuller development of the relation between the various kinds of points indicated by the condition

*dx o' the reader is referred to a paper by Mr Walton in the Cambridge Mathematical Journal, Vol. II. p. 155.

If the equation to the curve be put under the more symmetrical form

u = f (x, y) = 0, we easily obtain analytical conditions for distinguishing ben tween the three classes of double points indicated by the

0 condition 9 = , viz. true double points, points of osculation, and conjugate points. The condition 24 = involves the two,

da = 0, qu = 0.

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Proceeding to the differential of the second order, we find in consequence of the preceding condition

đỏ au , đau du, đou (d go

dae +2 dx dy dw* dy? \dx whence we find


d u

dy? Now for a true double point we must have two possible values for for a point of osculation we must have the two values equal; and for a conjugate point we must have the two values impossible. Hence we have the three conditions: I du 12 Pu\

>o for a true double point, (dx dyl Ilo

du 12 (du) du) . Goed) - 6 ) ) = 0 for a point of osculation, 1 ľu 12 (du) idul

<o for a conjugate point. \dx dy) (dx" ) (dyo ) If the point be more than double, it is necessary to proceed to higher differentiations, but the formula become too complicated to be of much use.

The second of the preceding conditions furnishes an easy demonstration of the following general property of curves of the third order. “ The three asymptotes of a curve of the third order being given, the locus of the points of osculation is the maximum ellipse which can be inscribed in the triangle formed by the asymptotes: the locus of the conjugate points is within, and of the double points without this ellipse."

If we refer a curve of the third order to two of its asymptotes as axes, their intersection being the origin, its equation must evidently be of the form,

ary + 2xy + cxy' = h.

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Therefore by the condition for a point of osculation

(a x + b + cy)? acry = 0, or aʻx? + a cxy + coyo + 2 abx + 2bcy + b2 = 0, which is the equation to an ellipse.

That this ellipse is the maximum ellipse inscribed in the triangle formed by the asymptotes is easily shown. The equations to the three asymptotes are

x = 0, y = 0, and ax + cy + 2b = 0.



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From the last it appears that the interceps of the axes

26 cut off by the third asymptote are - and -2. Also from the equation to the ellipse it appears that it touches the axes at distances and from the origin, or that the points of contact bisect these two sides of the triangle. If in the value of any derived from the equation to the ellipse we substitute the values - and - for æ and y,


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we find = --, which is the same as that derived from

doc the equation to the third asymptote, and as these values of u and y satisfy both the equation to the ellipse and that to the asymptote, it appears that the ellipse touches all the three sides of the triangle in their middle points, which by Chap. vii. Ex. 19, is the property of the maximum ellipse. The latter part of the theorem is too obvious to need demonstration. This proposition is due to Plucker, Journal de Mathématiques, (Liouville) Vol. 11. p. 11.

Points of Contrary Flexure or of Inflexion.

Ex. (1) The equation to the Witch of Agnesi is

æY = 2a (2ax – 2*)}; whence we find

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sa - h, when substituted for x, make any change sign, there are two points of contrary fexure corresponding to these values of x and y.

? and ? both become infinite when x = 0 and when dxdx w = 2a, but neither of those values gives a point of inflexion, since y is impossible when x is negative or greater than 2a. (2) The curve whose equation is

q? 3bxo + ay = 0
has a point of inflexion the co-ordinates of which are

X = b, y = :

(3) Let the equation to the curve be

a x? + by - c' = 0. There are two points of inflexion, the co-ordinates of the one being

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(1) Let the equation to the curve be

w" ax" + a' y = 0.
There are two points of inflexion corresponding to

a : 5 a
X = +

61: Y = 36:
(5) Let the cquation to the curve be

y = 0 + (0 – a)", where m and n are both odd.

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If ->1, X = a gives a point of inflexion, the tangent being parallel to the axis of x.

If <1, x = a gives a point of inflexion corresponding


? = 0, the tangent being perpendicular to the axis of x.

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