PL.-.'l-I A.'AT.YTI(‘.-L.] (1.) 3/=2.z:——1, as before; itis at once seen that this is a line; ' and taking it to be so, any two points, for instance, (0, —- 1) and (§, 0), deterniine the line. (2.) 3/=.z:"'. The equation shows that :3 may be positive or , negative, but that y is always positive, and has the same values for equal positive and iii_-gitivc values of .r: the curve y p:isses through the origin, and tlii-ougli the points (;i;1, 1). It is alieady known that the curve lies wholly above the axis of .73. To find its form in the neigh- bourhood of the origin, give CC a. small value, a'= :l:O'1 or ;l:O'01, then 3; is very much smaller, 0 --0'01 and 00001 in the two F." ,, uises respectively; this shows l~‘—" “' that the curve touches the axis of :6 at the origin. Moreover, an may be as large as we please, but when it is large, 3; is much larger ; for instance, ;c=10, y=100. The curve is a parabola (lig. :2). (3.) _i/=.‘c"‘. Here .c being positive 3/ is positive, but .1’ being nega- tive 3/ is also negative : the eiirve pisses through the origin, and also through the points (1, 1 and (— 1, — 1). Moreover. when .r' is small, -— 0'] for example, then not only is 1/, =0‘UO1, very much smaller than 9-, but it is also very much siiiallei‘ than y was for the last- iiieiitioned curve 3/=.7_-‘-’, that is, in the neighbourhood of the origin the present curve ap- proaches iiioi'e. closely the axis of an The axis of :7} is a tangent at the origin, but it is a tangent of a peculiar kind (a stationary or inllexional tangent), cutting the eurve at the origin, which is an intlexion. cubical parabola (fig. 3). _ _ (4.) 3/'-’=:c— 1.:c— 3.;::— 4. Hei'e Z/=0 f01'-73:1; =3. =4- “ 119_11' ei-er 93-1_.1:—3,a,--4 is positive, 3/ has two equal and opposite values ; but when .l'— 1.3: — 3.9: — 4, is negative, then 3/ is imagin- ary. In particular, for .1: less than 1, or between 3 and 4, 3/ 1S imagin- ar_v, but for 9: between 1 and 3, or greater than 4, 3/ has two values. It is clear that for 9: s« -mewliere between 1 and 3, 3/ will attain a iiiaxi- nnini, the values of 2: and 3/ maybe found ap- proximately by trial. The curve will consist of an oval and infinite branch, and it is easy to see that, as shown in fig. 4, the curve where it cuts the axis of w cuts ll at right angles. It FT 4 may be further remarked 15' ' that, as re increases from 4, the value of y will increase more and more rapidly ; for instance, a:=5, 3/"=8, a:=10, y2=378, &c., and it is easy to see that this implies that the curve has on the infinite l-raneh two intlexions as shown. (5.) y"’=:r—c.a:—b.a:—a, where a>b>c (that is, a nearer to + cc . c to — oo ). The curve has the same general form as in the last figiire, the oval extending between the limits :z‘.=c, :t=b, the infinite braneh commencing at the point x=a. (6.) y‘-’=(.2: — c)‘*(.c — a). Suppose that in the last-iiientioiied curve, y‘-’=n: — 0.3: — b..r — a, b gradually diiniuishes, and be- 3' comes ultimately = c. The iii- fiiiite branch (see fig. 5) changes its form, but not in a very marked manner, and it retains the two iiifiexioiis. The oval lies always between the values
- r=c. 9:=b, and therefore its
length continually diminishes; it is easy to see that its breadth will also eontinuall_v diminish; iiltiiiiately it shrinks up into a more point. The curve has thus a eon_]ugate or isolated Fig_ 5_ point, or aeiiode. For a direct verification observe that :c==c, 3/=0, so that (c, 0) is a point of the curve, but if x is either less than c, or between c and a, y" is negative, and y is imaginary. yl Fig. 3. The curve is the (/KB A K? CB GEOMETRY ‘ finite branch, which begins at ‘ a'=a. 409 (7.) 3/-’=(w—c) (9: -a)’. If in the same curve b gradually in- creases and becomes ultimately -=a, the oval and the infinite branch change each of them its form, the oval extending always between the values x=c, w=b, and thus continually approaching the iii- 3/ / I C A 3,, that finally, when b becomes . =a, the curve has the form F13‘ 6' shown in fig. 6, there being now a double point or node (eriiiiode) at A, and the intlexions on the infinite branch having disappeared. In the last four examples the curve is one of the cubical curves called the divergent parabolas: 4 is a mere iiumerical example of 5, and 6, 7, 8 are in Newton’s language the parabola cum orali, punclala, and notlata - respeetively. When ‘ (1, b, c are all equal, or the form is y‘~’=(.7:—c)-'*, we have a euspidal form, Newtoii's parabola cus- piclata, otherivise the semieubical parabola. (8.) As an example of a curve given by an im- plicit equation, suppose the equation is 9c”+3/”—3.z'y=0; this is a nodal cubic ‘ eiir'e, the node at the origin, and the axes touching the two branches i'e.<:pectively (fig. 7). An easy mode of tracing it is to express 9.‘, 3/ each of them in terms of a variable 0, = 1—:_v(:,:, 9 .7! = 1:97 ; but it is instructive to trace the curve directly from its equation. 5. It may he reniarked that the purely algebraical pro- ccss, which is in fact that employed in finding a differen- 1 The consideration of a few numerical examples, ‘ii-itli careful drawing, would show that the oval and the infinite branch as they approach sharpen out each towards the other (the two inflexioiis on the iii- finite braiieli coming always nearer to the point ((1., 0) ),—so 1,/. IL’ I tial coeflicient :,L: , if applied directly to the equation of the curve, determines the point consecutive to any given point of the curve, that is, the direction of the curve at such given point, or, what is t-he same thing, the direction of the tangent at that point. In fact, if a, ,8 are the coordinates of any point 011 a curve f(.r, 3/)=O, then writing in the equation of the curve x=a+/2, 3/= ,8+l-, and in the resulting equation f(a+71, ,8 +/.')=O (de- veloped in powers of Ii and Ir), omitting the tel’111f(o., ,8), which vanishes, and the terms containing the second and higher powers of 7:, Ir, we have a linear equation A}; + B]: = O, which deterniines the ratio of the increments /2, 1:. Of course, in the analytical development of the theory, we translate this into the notation of the difl"ei'eiitial calculus ; but t-he question presents itself, and is thus seen to be solvable, as soon as it is attempted to trace a curve from its equation. G'e0metrg/ is Dcscrz'12tz've, or Ilfetricul. 6. A geometrical proposition is either dcscr2'p1‘z'rn or metrical: in the former case it is altogether independent of the idea of magnitude (length, inclination, &c.) ; in the latter case it has reference to this idea. It is to be noticed that, although the method of coordinates seems to be by its inception essentially metrical, and we can hardly, except by metrical considerations, connect an equation with the curve which it represents (for iii- staiice, even assuming it to be known that an equation A.':.'+B_1/ + C = 0 represents a line, yet if it be asked what I line, the only form of answer is, that it is _the line cutting
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