526 



CURVE LINES. 



Plate 

 CCXXVII, 

 Fig. 11. 



Kg. 12. 



sections, we know from geometrical principles, (see 

 Geometry and Conic Sections); for a straight line 

 cannot cut another straight line in more than one 

 point, nor a conic section in more than two. 



32. Although, generally speaking, a straight line 

 may cut a line of any order in as many points as there 

 are units in the number expressing its order, yet this 

 will not always be the case ; the roots of an equation 

 may be impossible, and then they can have no geome- 

 trical expression. 



S3. As the roots of an equation become always im- 

 possible in pairs, so the intersections of a curve and its 

 ordinate must vanish in pairs, if any vanish. Let PM 

 (Fig. 11.) cut the curve in the points M and m, if it be 

 supposed to move parallel to itself, so as at last to 

 touch it in the point M', then the two points of inter- 

 section M, m go into one point of contact M'. The 

 line being supposed to continue its motion, it falls en- 

 tirely without the curve, and there is no contact. 



34. As all equations of an odd degree, viz. the third, 

 fifth, seventh, &c. have at least one real root, the equa- 

 tions of lines of the same orders will give at least one 

 real value of y for every value of x : Now, x may in- 

 crease indefinitely in both directions from the origin of 

 the co-ordinates ; therefore, such a curve will have at 

 least two infinite arcs. 



Again, as the roots of equations of even degrees, viz. 

 the second, fourth, <&c. may be possible only within 

 certain limits, so, in curves of these orders, the values of 

 x, Avhich give real values of y, may be confined within 

 certain limits, and hence the curve may be contained 

 within certain bounds, so as to have the figure of an oval. 



35. When two values of y are equal, the ordinate 

 either touches the curve, or meets it in what is called 

 a Punctum Duplex; two of its arcs intersecting each 

 other in that point ; or else some oval belonging to that 

 kind of curve becomes infinitely little at the top of the 

 ordinate, forming there a Punctum Conjugatum. 



If in the equation of the curve y be made — 0, the 

 roots of the equation by which x is determined will 

 give the distances of the points where the curve meets 

 the axis of the abscissa; from the origin. If two of these 

 roots are equal, the axis touches the curve, or passes 

 through a punctum duplex in the curve : When y — 0, 

 if one of the values of x then vanish, the curve in that 

 case passes through the origin; but if two vanish, the 

 axis either touches the curve, or the origin of the co- 

 ordinates is a. punctum duplex. 



36. In order to illustrate these observations, we shall 

 now shew, by particular examples, how the figure of a 

 curve may be determined from its equation. 



Let the equation of the curve be y' l — ax + ab. In 

 this case y— z±=*/(a.z-f ab). By giving particular 

 values to AP = x (Fig. 12.) and substituting for a and 

 6 their numeral values, we may find any number of 

 values of PMr=#, and thence any number of points 

 M, M', &c. in the curve: and as for every value of x, 

 y has two values, one positive and the other negative ; 

 corresponding to each, there will be two points M, m 

 at equal distances from the axis AX, and on opposite 

 sides of it. The greater x is taken, the greater is 

 a/ (ax -f- ab) z=y ; if x be supposed infinitely great, y is 

 also infinitely great, so that the curve has two infinite 

 arcs, which go off to an infinite distance from the axis 

 AX. If we suppose x —0, then y—z±=.\/ab; from 

 which it appears that y does not vanish, and therefore 

 the curve does not pass through A the origin of the 

 co-ordinates, but meets the axis YAj/ in two points 

 D, d, so that AD= \d=i/'a~b. 



Suppose now that P moves to the other side of A, so 



Curve 

 Lines. 



that x = AP' is to be accounted negative ; then P'M'= 

 y— ±y' '{ab — ax); here y has two values as before, 

 as long as x is less than b ; when x =z b, then y = ^■"Hr 

 z±=.*/(ab — ab) =0, so that the curve passes through B, a 

 point in A x, such, that AB=r&. If P be supposed to 

 move beyond B, so that x~^b, then ab — ax being ne- 

 gative, y = :±r y' {ab — ax) becomes imaginary, that is, 

 beyond B, there are no ordinates that meet the curve ; 

 and consequently, on that side, the curve is limited at 

 B. All this agrees with what is known by the theory 

 of the conic sections ; for the curve is evidently a para- 

 bola, whose vertex is B, and axis BAX, and the para- 

 meter of the axis =a. See Conic Sections. 



36. Let the equation of the curve be xy-{-ay-\-cy 



=.bc-\-bx: In this case, y= ~ ; and as y has on- 

 fit -4- C -j- x 



ly one value, corresponding to every value of x, the or- 

 dinate PM=« (Fie-. 13.) can meet the curve only in Plate 



* i c ' CCXXVIS 



one point : When x= 0, then y ss , so that the curve Fig. 13. 



does not pass through A, the origin : If x be supposed 

 to increase, then y increases, but never becomes equal 



to b, because y = b — , and a-Uc + x is always 



J o-f-c-j-x ' ' - . ■: 



greater than c-f-x. If x be supposed infinite, then the 



quantities a and c are to be accounted as nothing in re- 



x 

 spect of x; and in this case y — b — z=.b, from which it 



appears, that taking KD—b, and drawing DG paral- 

 lel to AX, it will be an Assymptote, and touch the curve 

 at an infinite distance. 



If * be now supposed negative, and AP' be taken on 



the other side of A, then shall y=b — , and if x 



J a-j-c — x 



be taken on that side, =c, then y ts — — 0, so that if 



" a 



AB=c, the curve must pass through B. 



If x become greater than c, then will c — x become 



negative, and the ordinate will be negative, and be on 



the other side of the axis, till x becomes equal to a -f- c, 



and then y = b— — , that is y is infinitely great, so 



that if A K be taken — a-\-c, the ordinate KL will be 

 an assymptote to the curve. 



If x be taken greater than fl-f-c, or AP" greater than 

 AK, then c — x and a -f c — x become both negative, and 



consequently y zz b becomes positive ; and since 



x — c is always greater than x — a — c, it follows that y 

 will be always greater than J or KG, and consequently 

 the rest of the curve lies in the angle FGH ; and as x 

 increases, since the ratio of x — c to x— a — c approaches 

 continually to a ratio of equality, it follows that PM 

 approaches to an equality with PN, and the curve ap- 

 proaches to its assymptote GH, on that side also. This 

 curve is the common hyperbola. 



37. The theory of curve surfaces, and of lines of double Curve 

 curvature, follows next in order that of plane curves. Surfaces, 

 This is a subject, however, of great extent ; and to en- 

 ter upon it at any considerable length would require 

 more room than the limits of our work will allow. We 

 shall, therefore, give a very brief sketch of some of the 

 principles of this branch of geometry. 



38. The position of any point in space is determined, 

 when we know the directions and the lengths of three 

 Straight fines, drawn from the point parallel to three 

 planes, and terminated by them. For greater simpli- 





