« PreviousContinue »
Or, if d = ac the semiaxis ; c = ch the semiconjugate ; p=0? •d the semiparameter ; x = ck the absciss counted from the centre ; and y = DK the ordinate as before. Then is AK = =d-x, and kb=d + x, and AK . KB = (d-30) x (d + x) d2
Then, by th. 2, d?:c2::d2-22: y', and days=c%d8-xo), or dy V (da x?), the equation of the curve. "
Or,d:p::d? - 32 :y>, and dy? = pid? - ), another form of the equation to the curve ; from which any one of the quantities may be found, when the rest are given.
2. For the Hyperbola.
Because the general property of the opposite hyperbolas, with respect to their abscisses and ordinates, is the same as that of the ellipse, therefore the process here is the very same as in the former case for the ellipse ; and the equation to the curve must come out the same also, with sometimes only the change of the sign of a letter or term from +' to -, or from - 10 +, becaäse here the abscisses lie beyond or without the transverse diameter, whereas they lie between or upon them in the ellipse. Thus, making the same notation for the whole diameter, conjugate, absciss, and ordinate, as at first in the ellipse ; then, the one absciss as being x, the other BK will be d + x, which in the ellipse was d X; so the sign of x must be changed in the general property and equation, by which it becomes d? :c2: : x (d + x):ya; hence day? cm (dx + x2) and dy = CV (dx+x), the equation of the
Or, using the parameter as before, it is, d:::(d + x): ys, or dy2 = (dx + 3%), another form of the equation to the curve.
Otherwise, by using the same letters d, c, th, for the halves of the diameters and parameter, and x for the absciss ck counted from the centre; then is ax = x-d, and BK = x+d, and the property d? : 0 :: (2 d) x (x + d): y’, gives daya =(x-d2), or dy=cv(x - do),where the signs of dr and x2 are changed from what they were in the ellipse.
Or again, using the semiparameter, d: :: ** - di: y, and dy?
(x2 - do) the equation of the curve. But for the conjugate hyperbola, as in the figure to theorem 3, the signs of both x® and d2 will be positive ; for the property in that theorem being cas: cao :: Do + CA? : Dea,
it is d? :( ::x? + d2 :y? = de", or d'y'=f* (r+ d ), and
=CV (x? + d?), the equation to the conjugate hyperbola. Or, as d:p::+ d2 :y, and dys = f (x' + d), also the equation to the same curve.
On the Equation to the Hyperbola between the Asymptotes.
= a, CB
Let CE and co be the two asymplotes to the hyperbola dvd, its vertex being F, and zr, bd, AF, BD ordinates parallel to the asymptotes. Pui af or EF and BD = y. Then, by theor. 28, AF
CB BD, or a2 xy, the equation to the hyperbola, when the abscisses and ordinates are taken parallel to the asymptotes.
3. For the Parabola.
If z denote any absciss beginning at the vertex, and y its ordinate, also the parameter. Then, by cor. theoremi, AK: «D::KD:, or x:y::y:p; hencc px = yi is the equation to the parabola.
4. For the Circle.
Because the circle is only a species of the ellipse, in which the two axes are equal to each other ; therefore, making the two diameters d and c equl in the foregoing equations to the ellipse, they become ya = dx — x2, when the absciss x begins at the vertex of the diameter : and ys = d'-rs, when the absciss begins at the centre.
In every one of these equations, we perceive that they rise to the 2d or quadratic degree, or to two dimensions ; which is also the number of points in which every one of these curves may be cut by a right line. Hence it is also that these four curves are said to be lines of the ad order. And these four are all the lines that are of that order, every other curve being of some higher, or having some higher equation, or may be cut in more points by a right line.
ELEMENTS OF ISOPERIMETRY.
Def. 1. When a variable quantity has its mutations reguo lated by a certain law, or confined within certain limits, it is called a maximum when it has reached the greatest magnitude it can possibly attain ; and, on the contrary, when it has arrived at the least possible magnitude, it is called a minimum.
Def. 2. Isoperimeters, or Isoperimetrical figures, are those which have equal perimeters.
Def. 3. The Locus of any point, or intersection, &c. is the right line or curve in which these are always situated.
The problem in which it is required to find, among figures of the same or of different kinds, those which within equal perimeters, shall comprehend the greatest surfaces, has long engaged the attention of mathematicians. Since the admirable invention of the method of Fluxions, this problem has been elegantly treated by some of the writers on that branch of analysis ; especially by Maclaurin and Simpson. A much more extensive problem was investigated at the time of " the war of problems," between the two brothers John and James Bernoulli : namely, “ To find, among all the isoperimetrical curves between given limits, such a curve, that constructing a second curve, the ordinates of which shall be functions of the ordinates or arcs of the former, the area of the second curve shall be a maximum or a minimun." While, however, the attention of mathematicians was drawn to the most abstruse inquiries connected with isoperimetry, the elements of the subject were lost sight of. Simpson was the first who called them back to this interesting branch of research, by giving in his neat little book of Geometry a chapter on the maxima and minima of geometrical quantities, and some of the simplest problems concerning isoperimeters. The next who treated this subject in an elementary manner was Simon Lhuillier, of Geneva, who in 1782, published his treatise De Relatione mutua Capacitatis et Terminorum Figurarum,
His principal object in the composition of that work was to supply the deficiency in this respect which he found in most of the elementary Courses, and to determine, with regard to both, the most usual surfaces and solids, those which possessed the minimum of contour with the samne capacity ; and, reciprocally, the maximum of capacity with the same boundary. M. Legendre has also considered the same subject in a manner somewhat different from either Simpson or Lhuillier, in his Elements de Géométrie. An elegant geome.
trical tract, on the same subject, was also given, by Dr. Horsley, in the Philos. Trans. vol. 75, for 1775 ; contained also in the New Abridgment, vol. 13, page 653. The chief propositions deduecd by these four geometers, together with a few additional propositions, are reduced into one system in the following theorems.
SECTION I. SURFACES.
Of all Triangles of the same Base, and whose Vertices fall
in a right Line given in Position, the one whose Perimeter is a Minimum is that whose sides are equally inclined to that Line.
Let all be the common base of a series of triangles ABC" ABC, &c. whose vertices c', c, fall in the right line Lu, given in position, then is the triangle of least
D perimeter that whose sides AC, BC, are inclined to the line um in equal angles.
M For, let bm be drawn from B, perpendicularly to lm, and produced till
B DU = Bm: join ad, and from the point c where ad cuts lm draw bc : also, from any other point cl, assumed in lm, draw c'a, c'B, c'd. Then the triangles DMC, BMC, having the angle dcm = angle ACL (th. 7 Geom.) = MCB (by hyp.) DMC = BMC, and DM = BM, and Mc common to both, have also bc = BC (th. I Geom.)
So also, we have c's = c'B. Hence ac + CB=AC + CD = AD, is less than ac' + c'd (theor. 10 Geom.), or than its equal ac' t c'B. And consequently, AB + BC + Ac is less than AB + Bc' tac'. Q. E. D.
Cor. 1. Of all triangles of the same base and the same altitude, or of all equal triangles of the same base, the isosceles triangle has the smallest perimeter.
For, the locus of the vertices of all triangles of the same altitude will be a right line im parallel to the base ; and when lm in the above figure becomes parallel to AB, since NCB = ACL, MCB = CBA (th, 12 Geom.), ACL = CAB; it follows that CAB = CBA, and consequently ac CB (th. 4 Geom.).
Cor. 2. Of all triangles of the same surface, that which has the minimum perimeter is equilateral.
for the triangle of the smallest perimeter, with the same surface, must be isosceles, whichever of the sides be considcred as base : therefore, the triangle of smallest perimeter has each two or each pair of its sides equal, and consequently it is equilateral
Cor. 3. Of all rectilinear figures, with a given magnitude and a given number of sides, that which has the smallest pe rimeter is equilateral
For so long as any two adjacent sides are not equal, we may draw a diagonal to become a base to those two sides, and then draw an isosceles triangle equal to the triangle so cut off, but of less perimeter: whence the corollary is manifest.
To illustrate the second corollary above, the student may proceed thus: assuming an isosceles triangle whose base is not equal to either of the two sides. and then, taking for a new base one of those sides of that triangle, he may construct another isosceles triangle equal to it, but a smaller perimeter. Afterwards, if the base and sides of this second isosceles triangle are not respectively equal, he may construct a third isosceles triangle equal to it, but of a still smaller perimeter : and so on, by performing these successive operations, he will find that all the triangles will approach nearer and nearer to an equilateral triangle,
Of all Triangles of the Same Base, and of Equal Perimeters,
the Isosceles Triangle has the Greatest Surface.
Let ABO. ABD, be two triangles of the same с base AB and with equal perimeters, of which
D the one abc is isosceles, the other is not : then the triangle ABC has a surface (or an altitude, greater than the surface (or than the altitude) of the triangle ABD. Draw co through D, parallel to AB, 10 A
B eut ce (drawn perpendicular to AB) in c': then it is to be demonstrated that ce is greater than c'e.
The triangles AC'B, ADB, are equal both in base and altitude; but the triangle ac'B is isosceles, while ADB is scalene : therefore the triangle Ac'B has a smaller perimeter than the triangle ADB (th. I cor. 1), or than ace (by hyp.) Consequently ac' Vou, I. Z zz