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Curves referred to Polar Coordinates,
155 Radius of curvature in terms of r and p
156 Chord of curvature through the pole
157 Radius of curvature in terms of r and a
161 Conditions for the concavity and convexity of the curve
On the Methods of tracing the forms of Curves from their Equations.
162 General principles and examples
165 Tangent and normal to the cycloid
167 Radius of curvature of the cycloid .
GENERAL PRINCIPLES AND ANALYTICAL APPLICATIONS.
1. If any two quantities are so connected that any variation in the magnitude of the one implies a corresponding variation in the magnitude of the other, either of these quantities is said to be a function of the other. Such a connection is expressed algebraically by means of an equation involving the symbols of the two quantities and any other symbols of invariable magnitudes. Thus x and y are functions the one of the other in the equation x2
- a) = و
(a – 22)!
where a and b are supposed to represent invariable magnitudes. 2. If y be expressed directly in terms of X, as in (2), it is
y said to be an explicit function of x; if it be merely connected
with x by an unsolved equation, as in (1), it is said to be an implicit function of x.
3. The connection between x and y may be expressed generally by an equation F(x, y) = 0 ...
:(3), where F(x, y) denotes any expression whatever involving x, y, and constants. These constants are usually called parameters, a term borrowed from the theory of conic sections, where the word parameter is used to denote a certain fixed line. If we wish to represent in the most general manner that y is an explicit function of x, we may write y = f (x)....
.(4), f(x) denoting any algebraical expression which involves x and parameters. The equation (4) is easily seen to be a particular form of the equation (3); for we may write it thus,
y-f(x) = 0, y - f (20) being merely a particular instance of the general form F(x, y).
4. Functions may be termed mathematical or empirical ; mathematical, if the functionality is established by definition ; empirical, if discovered by observation. As an instance of the latter functionality, let y denote the attraction of the Sun upon the Earth, and x the distance between these two masses; then it is known by observation that
te being a constant quantity: y is in this case an empirical function of x.
5. A function f(x) is said to be continuous when, as x increases continuously, f (x) passes continuously from one possible value to another through all intervening values: the function is said to be discontinuous whenever this condition is violated. Take for instance
2 - a
then, as x keeps increasing continuously from 0 to a value a - h, where h is a positive quantity less than any assignable magnitude, it is plain that y also keeps continuously changing
1 through every gradation of value from to 00: but when & changes from a - h to a + h, y leaps from – o to +60
, without passing through the intervening values. see that in this case y is generally a continuous function, but that it experiences a dissolution of continuity when x becomes equal to a.
If we take
(x - a)' then, although, when x = a, y assumes the value o, yet this value of x does not correspond to a discontinuous state of the function, since, as x passes from a - h to a + h, there is no gap in the range of values of y.
6. Suppose y = f(x) to be the equation to a curve; then, if the function f (x) is continuous for a certain range of values of x, every two points of the locus will be joined by a continuous curve: on the other hand, if there is a dissolution of continuity at any point, and if the function be possible before and after x has passed its critical value, there will be a gap between two points of the curve corresponding to consecutive values of x. Thus, in the instance of the curve
the asymptote, of which the equation is x = a, is touched at opposite ends by the curve for two consecutive values of x, one greater and the other less than a by an indefinitely small magnitude.
7. Functions of X, which are expressed by the ordinary signs of algebra and trigonometry, are usually continuous, if we disregard certain dissolutions of continuity corresponding to peculiar and detached values of x. There are however exceptions to this principle. For example, if y = (-a),