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but the differential coefficients are possible till we come to the third, showing that the impossible branch has a contact of the second order with the plane of the axes. (12) In the curve
a'y' – 2 aba'y – 2 = 0 there is a point of osculation at the origin, and one of the branches experiences an inflexion. Such a point is called one of oscu-inflexion. See fig. 37. (13) The curve
y + axi – b’xy= 0 has a ceratoid cusp at the origin and an inflexion in another branch at the same point. The cusp has the axis of x as tangent, and the inflected branch touches the axis of y. The form of the curve is that of the letter R. See fig. 38. (14) The curve
(y – c) = (x – a) (x – 6), a>h, has an oval between x = a and X = b. When x = a and y = C there is a point of osculation, the common tangent being parallel to the axis of x. See fig. 39. (15) The curve
(x? + y')} = 4 a' x’y has at the origin a quadruple point, a pair of branches touching both the axes. The form of the curve is best seen by transferring the equation to polar co-ordinates, when it becomes
zo = a sin 20.
The greater number of the preceding examples are taken from Cramer's work, Chap. x. and Chap. XII.
ON THE TRACING OF CURVES FROM THEIR EQUATIONS.
Curves referred to Rectangular Co-ordinates.
BEFORE proceeding to give examples of the application of analysis to determine the form of curves when their equations are given, I shall say a few words on the principles of the interpretation of symbols in analytical geometry, as a knowledge of these is requisite for the understanding of the views which I have adopted both in the preceding and in the following pages.
By the principles of the Geometry of Descartes, the position of a point in a plane is known when its distances from two axes Ox, Oy intersecting each other at right angles are known: and a curve is defined as a series of points for which there exists the same relation between the ordinate y and the abscissa x. This relation is expressed by means of an equation f(x, y) = 0 between X, y and constants, which is called the equation to the curve. If we assign a series of values to one of the two variables & and y, the corresponding values of the other can be found by means of the equation f (, y) = 0: now so long as we consider this only as an arithmetical equation, the only values of x and y which we can use are positive numbers. If we agree that the values of ware to represent lines measured from 0 (fig. 40) along Ox, and values of y lines measured from O along Oy, we can by means of the arithmetical values alone of w and y determine the positions of all points within the angle xOy. But the equation f(x, y) = 0 for any value of one variable will frequently give an expression for the other variable which is not arithmetical, such as – a or (-a'), or more generally (+a")". Now there is no necessity for interpreting these expressions which are uninterpretable in arithmetic; but it is clear that we shall gain an advantage in the generalization of our results if we are able to interpret these expressions in any way consistent with the original definition of the symbols employed. It was soon seen by the early cultivators of this geometry that the first of these expressions (-a) could receive the geometrical interpretation that, if a represented a line measured in one direction, (-a) represented the same length of line measured in the opposite direction. This extension of the interpretation of the symbols is of great importance, since it enables us to express by the one equation, f(x, y) = 0, the position of a point in all parts of the plane in which the axes Ox and Oy lie; and no curve is considered to be completely traced unless the negative, as well as the positive, values of the variables be taken into account. This however is merely a matter of convention, and we might, if it were thought proper, restrict ourselves to the positive values of the variables and confine the curve to the angle a Oy. If instead of interpreting (-a) to mean the measuring of the length a in a direction opposite to that originally taken, we use the more general definition that – a means that the line a is to be turned round through two right angles, we are led to the general interpretation of such an expression as (+ a"), viz. that the line a is to be turned round through the nth part of four right angles. This gives us a farther extension of the use of the equation f(x, y)=0); for, as the turning of a line through a given angle is not confined to any one plane, we are enabled to express by the equation to the curve the position of a point situate in any part of space. To explain this, let us suppose that for a value x = a, we obtain a value y = (+)ā b; this implies that the length b is to be measured not along the axis of y, but along a line inclined to it at an angle --- 27: but as the axes are supposed to remain perpendicular to each other, this angle must be taken in a plane perpendicular to that of the original axes. Hence, if there be a series of values of y all affected by the same
quantity (+)", they will give rise to a branch of the curve lying in a plane inclined at an angle *** 27 to the plane of the original axes. If for different values of the index of + change its value, the branch does not lie in one plane, but is a curve of double curvature.
This use of the interpretation of the symbol (+ a")ă has not been generally adopted, but it is quite as legitimate an extension as that of the negative values of the variables, and for the thorough understanding of the course of a curve it is quite as necessary. For all the ordinary purposes however of the equations to curves it is sufficient to use only the positive and negative values of the variables, and to these I shall restrict myself, only observing, that when such an expression as (-a*)$ occurs, it is not to be called imaginary, nor is the curve to be said therefore to have no existence for that value; but it is to be interpreted as indicating that the curve there leaves the plane of the axes, which for convenience I shall call the plane of reference.
The student who wishes for more information regarding the general interpretation of formulæ in Analytical Geometry is referred to a paper by the Abbé Buée in the Philosophical Transactions for 1806, to Mr Warren's Tract on the Geometrical Interpretation of Imaginary Quantities, and to the Cambridge Mathematical Journal, Vol. 1. p. 259, and Vol. 1. p. 103 and p. 155: the last two papers being by Mr Walton.
When we proceed to trace a curve from its equation it is advisable in the first place to solve the equation with respect to one or other of the variables, if the solution be in a form which enables us to determine readily its value for different values of the other variable. After that we may proceed in the following way.
1. If y be the variable which is expressed in terms of x, assign to x all positive values from 0 to , marking those which make y = 0, y = 0, or y impossible. The first gives the points where the curve cuts the axis of x, the second gives the infinite branches, and the third, showing where the curve quits the plane of reference, gives the limits of the curve in that plane.
2. Assign to x all negative values from 0 to co, proceeding as in the case of the positive values of a. In both cases attend to both the positive and negative values of y, so as to obtain the branches on both sides of the line of abscissæ.
3. Find whether the curve have asymptotes, and determine them if they exist.
4. Find the value of ", and thence deduce the maximum and minimum points of the curve, and the angles at which the curve cuts the axes.
5. Find the value of , and thence deduce the nature of the curvature of the different branches, and the points of contrary flexure if such exist.
6. Determine the existence and nature of the singular points by the usual rules. Ex. 1. Let the equation to be discussed be
213 – a
From its form we see at once that there are always for each value of x two values of y equal but of opposite signs; hence the curve is symmetrical with regard to the axis of x.
Let « be positive ; when x is between 0 and a, y is impossible, and the curve does not exist in the plane of reference: when x = 0, y = 0: when x > a, y is possible, and increases without limit as x so increases.
Let x be negative; when a is between 0 and b, y is impossible, and there is no branch in the plane of reference: when x = b, y is infinite: when x >b, y increases without limit as x so increases. Hence it appears that the curve bas six infinite branches.
Since x = -b makes y infinite, the ordinate at that point is an asymptote. Also since