41. We have seen in art. 25 that, in figs. 1, 2, 3, 4, M'Q is = p. Ax, that is

that is to say, =

that is to say that the trigonometrical tangent of the angle MBP, included between the tangent MB of a curve and the axis of the abscisses, is equal to dy p= dx' +p, in figs. 1 and 2, when of the two adjacent angles about B, the acute angle MBP, is turned towards the positive side of the abscisses; and = —p, in figs. 3 and 4, when the acute angle MBP is turned towards the negative side of the abscisses.

Now, this angle varies with the point of contact; if we designate the cody ordinates of certain points of contact by x' and y', we must in dx put x' and y' instead of x and y, to obtain the trigonometrical tangent for this determinate point of contact, which will be expressed by

dy' da

or p'.

But it is to be observed that of these two quantities x' and y', x' only can be taken at pleasure, and the corresponding value of y' must be sought by the primitive equation of the curve.

To elucidate the preceding, let it be proposed to seek the trigonometrical tangent of the circle for the co-ordinates of the point of contact x' and y.

The equation of the circle referred to the centre being y = (r2

- x2), it fol

lows that y' = (r2 — x2); moreover p being =

the tri

Taking y' =

gonometrical tangent required will be = (r2 = x2) 3 positively, that is, considering the part of the circle above the diameter, in which the ordinates are positive (art. 25), we see by the sign of the preceding expression, on which side of the origin the co-ordinates of the tangent meet the axis of the abscisses; thus, a' being positive, the tangent meets the axis of the abscisses on the positive side of the abscisses, as in figs. 3 and 4 ; but x' being negative, as in figs. 1 and 2, it meets it on the negative side.

Reciprocally, when in a given curve is required the point of contact where the tangent with the axis of the abscisses includes an angle whose trigonometrical tangent is = α, the co-ordinates of the point of contact a' and y' must be deterdy' mined by the equation

= a, conjointly with the given equation of the curve.

Suppose, for example, that the tangent of the circle makes with the axis of the abscisses an angle = 60°, whose trigonometrical tangent is = +3, we have

the two equations

y' = (r2 — 2′2)3, and dy

It hence results that

r

da

r

x' = — . ± 33, and y' = + 1⁄2 (if y' is taken positively).

2

dy'

If we make

= 0, we shall obtain the co-ordinates of the point of contact, dx' where the tangent with the axis of the abscisses, makes an angle the tangent is parallel to the axis of the abscisses.

If, on the contrary, we make

dy'
dx'

==

= 0, or where

∞ (a condition which is satisfied by mak

ing the denominator in the value of dy equal to zero) the corresponding angle

is =

dx'

90°, that is, the tangent is perpendicular to the axis of the abscisses. For

= 0 indicates a maximum or a minimum (art. 37); therefore, since from

cates a maximum or a minimum, but in reference to the absciss x'. ness of these two propositions will appear clearly from the inspection of a figure, for example, of a circle.

42. We may easily find the sub-tangent BP (fig. 1, 2, 3, 4) by the preceding method. In the triangle MPB, we have

and, according to art. 41, being positive in figs. 1 and 2, and negative in figs.

3 and 4, the expression for the subtangent BP will also be positive in figs. 1 and 2, and negative in figs. 3 and 4.

In the same manner the position of the subtangent may at once be determined by the sign by which its expression is affected.

We find readily by the right-angled triangles BMR and PMR (figs. 1, 2, 3, 4) that the sub-normal PR is equal to

Its position, as well as that of the subtangent, may be ascertained by the sign of its expression. The length of the tangent MB and that of the normal MR may at once be deduced from the value of the sub-tangent and that of the sub-normal.

43. It may be well to recal to the reader, in a few words, some propositions of the elementary Analytical Geometry, concerning the equation of the right line.

The most general equation of the right line is

y = ax + b,

in which x and y denote the co-ordinates, and a and b constant (though indetermined) quantities, so that b represents the ordinate at the origin of the coordinates, and a the trigonometrical tangent of the angle comprehended between the right line and the axis of the abscisses, supposing that, as usual, rectangular co-ordinates have been adopted. The sign a is positive when, of the two angles found by the line and the axis of the abscisses, that which is acute is towards the side of the positive abscisses; which takes place with the tangent BM, which here represents the right line in general, in figs. 1 and 2, when the angle MBP < 90°.

On the contrary, a is negative when of the two angles, that which is obtuse is towards the side of the positive abscisses, which takes place with the tangent BM in figs. 3 and 4. Moreover, for the tangent BM, the point of intersection B, in figs. 1 and 2 is on the negative side, and in figs. 3 and 4 on the positive side of the origin A; therefore the ordinate is positive in all the four figures, because the ordinates are taken in a positive sense above the axis of the abscisses (art. 25). In the contrary case, the ordinate, or b in the general equation of the right line is negative. In this manner the reader may, without difficulty, trace and project the four following forms.

We will now consider some of these determinations.

Let it be proposed to find the equation of a right line, which, including with the axis of the abscisses an angle whose trigonometrical tangent = a', passes by a point whose co-ordinates are a and ß.

Besides the general equation y = ax + b, we then have also, for the given point, ẞ = aa + b;

eliminating b from the two equations, we obtain - 3 y = a (x a). Moreover, because of the given angle, we have for the equation sought,

and it is obvious that the letter b of the general equation is represented by the expression ẞ a'a, the sign of which remains still to be found, in order to determine the corresponding figure of the line according to the four particular forms given above.

In a similar manner we obtain for the right line which passes through two points whose co-ordinates are a, ß, and a', B', the following equation :

Finally, let it be proposed to find the equation of the right line perpendicular to another whose equation is y = a'x + b'.

The required line, in consequence of being perpendicular to the axis of the abscisses, forms an angle whose trigonometrical tangent. Moreover, the position of the required line belonging to the two latter of the four forms above, we have for the required equation

If moreover, the line sought passes through a point whose co-ordinates are a and ß, we have for its equation

44. Hence it is easy to find the equations of the tangent and of the normal for any curve.

The tangent to a point whose co-ordinates are a', y', and including with the dy' axis of the abscisses an angle whose trigonometrical tangent = (41) has for dx' equation (43)

Moreover, the normal being perpendicular to the tangent and passing through a point whose co-ordinates are x', y', has for equation (43)

Example. For the circle whose equation y2 + x2 = r2, we have, as has been already shown,

is

=

The absence of the constant term in this equation shews that the ordinate 0 (43) at the origin of the co-ordinates, which is here the centre, and that consequently the normal passes through the centre, that is, it is one of the radii. For another example, let there be given the equation

y2 = 2mx + nx2,

a general equation of all the sections of the cone referred to the summit and to

dy
da'

the axis. It is proposed to find and then the equation of the tangent and of

the normal.

This is left as an exercise for the student.

45. We will now return to the expression considered in No. 25,

Ay = p. Ax + ¥ . Ax2.

We have seen, in the above article, that is positive in figs. 1 and 3, where the curve turns its convexity towards the axis of the abscisses; negative in figs. 2 and 4, where it turns its concavity towards the axis of the abscisses. Now, we have (17)

and, if Ar is taken sufficiently small, will be positive or negative, according as q is positive or negative. Hence we see that the curve will in any point present its convexity or its concavity towards the axis of the abscisses, according as the differential coefficient of the second order q, obtained from the equation of the

7

curve, for the corresponding value of the absciss x, is positive or negative. It follows at the same time, that for a maximum in which p = 0 and q is negative (36) the curve necessarily turns its concavity towards the axis of the abscisses, and that for a minimum in which p = 0 and q is positive, the curve necessarily turns its convexity towards the axis of the abscisses: which will at once appear from the inspection of a figure *.

It is evident, both from the preceding and from art. 41, that in order to ascertain the position of a curve with respect to the axis of the abscisses in a given point, the signs taken at that point by p and q must be examined.

If q change its sign immediately before and after a given point, it may be concluded that on one of the sides of this point the curve is convex, and on the other concave towards the axis of the abscisses; such a point is called a point of inflexion. It is evident that in this point, in which its sign passes to the opposite, q either vanishes or is infinite, the latter circumstance taking place if q is a fraction whose denominator vanishes for this point.

In order more clearly to understand how a quantity passes from positive to negative, through the infinite, it will be well to call to mind that in Trigonometry, r sin. x which expresses the length of the tangent of an angle x, COS. x

the quantity

becomes infinite when cos. x = 0, or when the tangent ceasing to be positive, begins to become negative.

M

PO P

But even when q becomes, for a point of inflexion, either 0, or = the reciprocal of the proposition does not necessarily obtain, and there does not always exist a point of inflexion for q = 0, or = ∞ ; for the equation of the curve represented in the margin, which at A, has a point of regression, must necessarily contain a radical of an even degree. Thus only can ordinates become imaginary, and the existence of the curve on one of the sides of this point, impossible. But in equations of this kind, the first differential coefficients become = o for that value of a which makes the radical vanish, and the consecutive coefficients become∞: as has been already shewn, art. 23 and 26. In what follows, it will therefore be observed: if for a certain value of x, q is = 0 or = ∞, the primitive equation must be examined, whether it indicates the existence of the curve on one side only of this point, or on both sides at the same time; it is only in the latter case that the point examined can be a point of inflexion, while in the former case it may be a point of regression.

It often happens that for some values of x, for which p = 0 or = ∞, q also becomes = 0 or = ∞, and yet neither of these two points exist, which will be examined further on. For this reason, and because some values of æ, in which both p and q become = 0 or = ∞, do not correspond with any ordinate (as the primitive equation then shows) that it is absolutely necessary to consider the expressions of y, p and q, conjointly. But all this requires a more lengthened analysis than can here be given to it f.

46. Let us now consider two curves whose equations are

y = f (x) and y = p (x).

*For the principles employed in the determination of asymptotes, see page 241.

It is very elegantly treated, with various kindred inquiries, in Hind's Differential Calculus.