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can be suggested only from the particular form of the equation in view. By luppo fing x to have certain relations to the known quantities, the values of y may

become more simple, and the equation may be reduced to such a form as to fhew the direction of the curve, and some of its obvious properties.

The following general observations may

also be laid down.

1. If in any case a value of y vanishes, then the curve meets the base in a point determined by the corresponding value of x. Hence, by putting y=0, the roots of the equation, which in that situation are values

X will give the distances on the base from the point assumed as the beginning of x, at which the curve meets it.

of *,

X, Y

2. If at a particular value of becomes infinite, the curve has an infinite arc, and the ordinate at that point becomes an asymptote.

3. If, when x becomes infinitely great, y vanishes, the bale becomes an alymptote.

4. If any value of y becomes impossible,

j then so many intersections of the ordinate and curve vanish. If at any value of x all the values of y become impossible, the ordinate does not there meet the curve.

5. If two values of y become equal, and have the same sign, the ordinate in that fituation either touches the curve, or passes through an intersection of two of its branches, which is called a punctum duplex, or through an oval become infinitely little, called a punctum conjugatum.

In like manner is a puntum triplex, &c. to be determined.

The following example will illustrate this doctrine.

Let the equation be aya—xy?=x+bx2:

*3 +6x2 Therefore, y2= and y=+

x3 + bx2

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Let AB be assumed as a base on which the abfciffes are to be taken from A, and the ordinates perpendicular to it.


Since the two values of

y are equal, but have opposite signs; PM, and Pm v hich represent them, must be taken equal to each other on opposite sides of AB; and it is plain that the parts of the curve on the two sides of AB, must be every way similar and equal. If x is made equal to a, then y=x/ *+

+b which is an algebraical expression for infinity; therefore, if AC is taken equal to a, the perpendicular CD will become an asymptote to the curve, which will have two infinite arcs (Obf. 2.). If x is greater tnan a, the quantity under the radicał sign becomes negative, and the values of y are im

: possible ; that is, no part of the curve lies

i beyond CD (4.).

Both branches of the curve pass through A, since j=o, when x =o (1.). Let x be



the values

negative, and y=+* /

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of y will be possible, if x is not greater than b; but, if x=b, then y=0, and if x is greater than b, the values of y become impossible ; that is, if the absciss AP be taken to the left of A, and less than 6, there will


be two real equal values of


viz. PM, Pm on the opposite sides as before; if AE be taken equal to b the curve will pass through E, and no part of it is beyond E (1. and 4.).

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The portion between A and E is called a Nodus.

If y be put =0, then the values of x are 0, 0, —b. That is, the curve passes twice through A, or A is a punctum duplex, and it passes also through E as before (1.).

The mechanical description of curves, mentioned in the beginning of this section, may be illustrated by the preceding example. For this purpose, let any numeral



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values of a and b be assumed ; and if fucceffive numeral values of x be inserted, corresponding numeral values of y will be obtained, by which so many points in the curve may be constructed. Let AC=a=10; AE=b=12; and, first,

x+6 let x=I, then y=+x

W +1.2 nearly, which gives the length of the ordinates when the absciss is 1 ; and in the same manner are the ordinates to be found when x is 2, 3, or any other number. Thus, if x=6, then y=+6x118=12.73

, 6=6X -= nearly; and if AP be taken from the scale of equal parts (according to which AB and AE are supposed to be laid down) and equal to 6, then PM, Pm, being taken from the same scale, each 'equal to 12.73, will give the points of the curve M, m. In like manner, if x=-9, y=+9/3

=, 3.58, nearly, and if AP=9, then PM, Pm being taken from the same scale equal to 3.58, will give the points M, m. In the



二十 19

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