can be fuggested only from the particular form of the equation in view. By suppofing x to have certain relations to the known quantities, the values of y may become more fimple, and the equation may be reduced to fuch a form as to fhew the direction of the curve, and fome of its obvious properties. The following general obfervations may alfo be laid down. 1. If in any cafe a value of y vanishes, then the curve meets the bafe in a point determined by the corresponding value of x. Hence, by putting yo, the roots of the equation, which in that fituation are values of X, will give the diftances on the base from the point affumed as the beginning of x, at which the curve meets it. 2. If at a particular value of x, y becomes infinite, the curve has an infinite arc, and the ordinate at that point becomes an asymp tote. 3. If, when x becomes infinitely great, y vanishes, the baie becomes an afymptote. 4. 4. If any value of y becomes impoffible, then fo many interfections of the ordinate and curve vanish. If at any value of x all the values of y become impoffible, 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 situation either touches the curve, or paffes through an interfection 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 punctum triplex, &c. to be determined. The following example will illuftrate this doctrine. Let the equation be ay2—xy2=x3+bx2: Let AB be affumed as a base on which the abfciffes are to be taken from A, and the ordinates perpendicular to it. Since x+6 Since the two values of y are equal, but have oppofite figns; PM, and Pm which represent them, must be taken equal to each other on oppofite fides of AB; and it is plain that the parts of the curve on the two fides of AB, must be every way fimilar and equal. +6 If x is made equal to a, then y=x; which is an algebraical expreffion for infinity; therefore, if AC is taken equal to a, the perpendicular CD will become an afymptote to the curve, which will have two infinite arcs (Obf. 2.). If x is greater than a, the quantity under the radical fign becomes negative, and the values of y are impoffible; that is, no part of the curve lies beyond CD (4.). Both branches of the curve pafs through A, fince j=0, when x=0 (1.). Let x be of y will be poffible, if x is not greater than b; but, if x=b, then y=o, greater than b, the values of y and if x is become im poffible; that is, if the abfcifs AP be taken to the left of A, and lefs than 6, there will be be two real equal values of y, viz. PM, Pm on the oppofite fides 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.). The portion between A and E is called a Nodus. If y be put o, then the values of x are 0, 0, -b. -b. That is, the curve paffes twice through A, or A is a punctum duplex, and it paffes alfo through E as before (1.). The mechanical defcription of curves, mentioned in the beginning of this fection, may be illuftrated by the preceding example. For this purpose, let any numeral values values of a and b be affumed; and if fucceffive numeral values of x be inferted, corresponding numeral values of y will be obtained, by which fo many points in the curve may be conftructed. Let AC=a=10; AE=b=12; and, first, 1.2 nearly, which gives the length of the ordinates when the absciss is 1; and in the fame manner are the ordinates to be found when x is 2, 3, or any other number. ! Thus, if x=6, then y=±6x18 =12.73 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. like manner, if x In 3 x=9, y=±9 19 + 3.58, nearly, and if AP=9, then PM, Pm being taken from the fame fcale 3.58, will give the points M, m. equal to In the fame |