(TKYK. 



i ru\ K. 



know any curve which contain* them all. It would appear as if our 

 >ure contained several curve*, but it must be remembered that in the 

 algebraical aenae many currea exist with branchM completely uncon- 

 nected, but all described under one law. 



The several parts of the preceding are of continual occurrence ; the 

 following are the names' and references : 



(1). Points of inflexion or contrary flexure. [FLEXURE, CONTRARY.] 

 MULTIPLE FOISTS, double, triple, &c., according to the number 

 of times the curve paimrn through them. 



(8). CUSPS : the terms are hardly sufficiently well settled to enable 

 us to say whether the 3* is to be considered a double point, a triple 

 l >iiit. or not a multiple point at all. 



(4). It is customary to call any part of the curve which encloses 

 space an OVAL, though, according to the common meaning of the terui, 

 there is no oval in our diagram except 4t. Of 4* we hardly know 

 whether it would be called an oval or not. 



(6). Conjugate points [CONJUGATE]; when of a general law of 

 description which gives ovals, a particular case is taken in which an 

 oval disappears, it generally leaves behind it, so to speak, a single point 

 which is included under the equation to the curve, but has no con- 

 tiguous points. We should propose to call these points eranttcent oral. 

 1 INVOLUTE, EVOLUTE, CAUSTIC, CONTACT, TANUEST, ARC, AREA, 

 ASYMPTOTE, MAXIMA and MINIMA, &c.] 



Some remarks on a few of the curves in the preceding list of named 

 are given here. 



6. Sftnicubical fiarabola. The term parabola has been extended to 



mean any curve having an equation of the form y ax', where m is 



positive. Thus y = <u? is the cubical parabola, and y=<u is the semi- 

 cubical parabola. The seinicubical parabola is the evolute of the common 

 parabola. The term hyperbola luua been extended to any case in which 

 m i* negative. 



8. Trittctrix. See this curve described in TRISECTION, and figured 

 in TROCHOIDAL CCKVES. 



11 and 12. For the Companion to the Cycloid, see CYCLOID. The 

 harmonic curve, so called because it, or a portion of it, is one of the 

 simplest forms assumed by a vibrating string, has for its equation the 

 relation between x and y implied in 



x=a6 y = b(l cos*). 



12 to 17. These curves are all described in TROCHOIDAL CURVES. 

 20 to 24. See SPIRAL. 



30. Oralt of Cassini. Dominic Cassini (see James Caasini's ' Astro- 

 nomy,' vol. L p. 149) proposed, as a better representation of the 

 planetary motion than the ellipse, a curve in which the rectangle of the 

 distances from any point in it to two fixed foci is constant. This curve 

 is of the fourth order, and may be one oval, a lemniscate form, or two 

 separate ovals, according to the ratio of the given rectangle to the 

 square on the distance between the foci. It is hardly necessary to say 

 that Cassini was wrong in his application of this curve : but the 

 celebrity of his name has kept his oval or ovals among the curves 

 which serve for exercise to beginners. 



31. Watt'i Parallel Motion Curre. The end of the beam of a 

 steam-engine describes an arc of a circle ; but it is required that the 

 end of the piston-rod, which must be in some way attached to it, 

 should describe a straight line, or a curve which is very nearly a 

 straight line. Now at and near a |>oint of contrary flexure the arc 

 of a curve is very nearly straight. If two rods revolve round fixed 

 piviits, their ends being connected by a third rod, the middle point oi 

 that third rod will describe a cun-e which has a point of contrary 

 flexure ; and if the lengths of the rods be properly taken, and if one 

 of the rods be the beam of the steam-engine, the reciprocating rotatory 

 motion of the beam may be made to communicate what is practical!] 

 a reciprocating rectilinear (improperly called parallel) motion to thi 

 middle point of the third rod, on account of the arc described by that 

 middle point containing the point of contrary flexure. A piston there 

 fore may be attached to the middle point of the third rod, and the 

 requisite condition is practically satisfied. The apparatus described in 

 the article STEAM-ESOI.XE, and in Lardner's ' Steam-engine,' consists in 

 the addition of a pair of rods which with the first and third compose a 



rallelogram moveable at all iU joints. This is intended to furnish 

 he same reciprocating rectilinear motion to a second piston placed at 

 the new joint, which is nothing to our present purpose. Watt's 

 larallel motion curve must be defined as the locus of the middle point 

 of a straight line of given length, the ends of which describe arcs of 

 circles. 



The following considerations on the definition of the word curre will 

 w useful to the young analyst : 



Geometrically speaking, a curve is described by a point which moves 

 according to one uniform law, and does not describe a straight line. An 

 oval, for instance, formed by arcs of circles in the manner described in 

 xx>ks on mensuration, is not a curve, but a junction of several arcs of 

 different curves : and even the two branches of an hyperbola cannot be 

 said, in this primitive view of the subject, to be anything but two 

 curves ; accordingly they were called opposite hyperbolas. 



But, algebraically speaking, the meaning of the word curve is much 

 extended. Every curve, just defined, has one permanent equation 

 connecting the abscissa and ordinate of every point in it. The con- 

 verse is made true by extension of the definition of the word 

 All the points whose co-ordinates satisfy an equation constitute the 

 curve to which the equation is said to belong. Let x and y be the 

 abscissa and ordinate of a point ; and 9(7, ;/) = 0, the equation which is 

 to connect them. If this equation be satisfied when x = a and y = 6, 

 the point whose co-ordinates are a and b is considered in every case as 

 belonging to the curve whose equation is $>(.r, y) = 0. The consequence 

 is that a curve, or what is so called in algebra, may be either the 

 simple curve of geometry or a collection of such curves formed under 

 algebraical modifications of one law or a collection of such curves not 

 even connected by one law or an isolated point or a collection of 

 isolated points at finite distances from one another or a r ;. ii..n f 

 isolated points infinitely near to one another, but which nevertheless 

 cannot be considered as forming one continuous branch of an ordinary 

 curve. Or it may be any combination of two or more of these. We 

 give some instances, as follows : 



1. The equation y m*j? = a* belongs to both the branches of on 

 liyperbola. If a change, m remaining constant, the hyperbola becomes 

 a different hyperbola, with the same asymptotes as before. But if 

 (i O, the equation belongs simply to those two asymptotes, that 

 is, to two distinct and independent straight lines. In like manner ; IP 

 equation 



belongs to a complicated curve, with various branches. But if o = l, 

 this curve resolves itself into two distinct curves, the straight line 

 ar + y=l, and the circle x* + y t = \. For the equation can then !* 

 reduced to 



which is satisfied whenever cither a? + y* 1 = or x + y 1 = is 

 satisfied. If a, instead of being =1, in only very near to 1, the rune 

 approaches very near to the circle just mentioned throughout, and, 

 except for great values of r, also to the straight line. 

 L The equation 



(x a 



belongs to a distinct pair of straight lines when m is negative. I'.ut 

 when m is positive, it belongs to nothing but the point whose co- 

 ordinates are a and b. 

 8. The equation 



y = ax* + sin a;.** 



belongs to a branch with an infinite number of intersecting convolutions, 

 when x is positive : but when .r is negative, it belongs to nothing but 

 an infinite number of isolated points, situated at finite distances from 

 each other on a parabola. 



4. If y=a", a being positive, there is a continuous branch in which 

 there is a positive ordinato for all values of .T. There can be no 

 negative ordinate except for values of .r which have, when in their 

 lowest terms, odd numerators and even denominators. But between 

 any two values of x can be interposed on infinite number of Midi 

 fractions. There is then a branch of the curve which, though it may 

 be said to contain an infinite number of points, infinitely near to: 

 cannot be called continuous. For there is no negative value of y w hen 

 a: is a fraction which has an even numerator and an odd denominator. 

 These pointed branches, as they have been called, havo only been 

 recently considered. Their adnuaaion depends entirely upon definition. 

 Those who would restrict the meaning of the word curve to soni> 

 like its ancient signification, will reject them : but those who wish to 

 look upon a curve as a tabulation in space of every possible v.ili 

 and y which will satisfy a given equation, must admit them. Ti 

 hints hero given on the extension of the word curve are meant to 

 excite the curiosity of those who have not been accustomed to look at 

 geometrical interpretation as subordinate to algebra. 



A distinction ought to be Iwcen /;/<' 



geometrical algebra. In algebraic geometry, in wli ich algebra is brought 

 to the assistance of geometry, the old notion of the meaning <>) 

 might be retained. But in geometrical algebra, in which geometry is 

 used in illustration of algebra, all the extensions which we have men- 



