Paraboloid

“ A paraboloidal reflecting surface as in Fig. 65 has the property of bringing to a single point focus all rays parallel to its axis while [Pg 94] quite failing of uniting rays proceeding from any point on its axis, since its curvature is changing all the way out from vertex to periphery. Here the parallel rays a, a, a, a meeting the surface are reflected to the focus F, while in a perfectly symmetrical way the prolongation of these rays a′, a′, a′, a′ if incident on the convex surface of the paraboloid would be reflected in R, R′, R′′ R′′′ just as if they proceeded from the same focus F.
Fig. ”

“ The parabola falls away toward the periphery and hence pushes outward the marginal rays. But it is of relatively sharper curvature near the center and pulls in the central to meet the marginal portion. In the actual construction of parabolic mirrors one always starts with a sphere which is easy to test for precision of figure at its center of curvature. Then the surface may be modified into a paraboloid lessening the curvature towards the periphery, or by increasing the curvature toward the center starting in this case with a sphere of a bit longer radius as in Fig. ”

“ There was only one parabola, and that was formed when the side of the cone was parallel to the plane of section, that is when the falling grease just touched the edge of the candlestick; there is only one bubble with no pressure, the catenoid, and this is drawn by rolling the parabola. As the cone is gradually inclined more, so the sections become at first long ellipses, which gradually become more and more round until a circle is reached, after which they become more and more narrow until a line is reached. ”

“ As each parabola corresponds to a different kind of charged particle this shows that there are only a limited number of different kinds of particles in the discharge tube. The sharpness of the lines shows that all the particles of the same chemical element have to a very high degree of accuracy the same mass. If the atoms of hydrogen, for example, differed appreciably in mass it would not be possible to get on the photographic plates parabolas as sharply defined as those which can be obtained when the tube through the cathode has a very fine bore. ”

“ When exceedingly large electric fields are used it can be seen, however, that the head of the mercury parabola is distinctly displaced, and on measuring the amount of the de-flection it is found to be one-eighth of the normal displacement of the heads of the parabolas corresponding to the other elements. ”

“ As one parabola after another was passed, positive electricity would come abruptly through the slit, and the amount of the charge would be a measure of the number of particles passing through the slit. If instead of moving the parabolic slit we keep the slit fixed and gradually increase the magnetic field used to deflect the particles, we shall in this way drive one parabola after another on to the slit, beginning with the parabola due to the hydrogen atom and ending with that due to the mercury one, and the charges passing the slits will be proportional to the number of particles. ”

“ I called one of the bubble forms, if you remember, by its name, catenoid; this is produced when there is no pressure. The dotted curve in the second figure B is this one; and to show that this catenary can be so drawn, I shall roll upon a straight edge a board made into the form of the corresponding section, which is called a parabola, and let the chalk at its focus draw its curve upon the black board. There is the curve, and it is as I said, exactly the curve that a chain makes when hung by its two ends. Now that a chain is so hung you see that it exactly lies over the chalk line. ”

“ It is to be remarked that the classification mixes together non-singular and singular curves, in fact, the five kinds presently referred to: thus the hyperbolas and the divergent parabolas include curves of every kind, the separation being made in the species; the hyperbolisms of the hyperbola and ellipse, and the trident curve, are nodal; the hyperbolisms of the parabola, and the cubical parabola, are cuspidal. The divergent parabolas are of five species which respectively belong to and determine the five kinds of cubic curves ”

“ This is sometimes termed the campaniform (or bell-shaped) parabola. If the two greater roots are equal the equation is (in which a < b) and the curve assumes the form shown in fig. 6, and is known as the nodated parabola. Finally, if all the roots are equal, the equation becomes this curve is the cuspidal or semi-cubical parabola (fig. 7) . This curve, which is sometimes termed the Neilian parabola after William Neil (1637—1670) , is the evolute of the ordinary parabola, and is especially interesting as being the first curve to be rectified. ”

“ The involution on this axis has its centre at infinity. One focus is therefore at infinity, the one focus only is finite. A parabola has only one focus.
§ 85. If through any point P (fig. 34) on a conic the tangent PT and the normal PN (i.e. the perpendicular to the tangent through the point of contact) be drawn, these will be conjugate lines with regard to the conic, and at right angles to each other. They will therefore cut the principal axis in two points, which are conjugate in the involution considered in § 84; hence they are harmonic conjugates with regard to the foci. ”

“ If A represents a particle with a threefold charge there will be another stronger parabola B for which e/m has three times the value corresponding to the parabola A, and B will have a prolongation towards the vertical axis extending to one third of the normal distance. ”

“ The sand, shot backward under the creature, clears the arch of its hind legs, trickling like a liquid in a continuous thread, describing a parabola and falling some eight inches away. This dusty jet, constantly fed for five or ten minutes, is enough to show with what dizzy rapidity the tools are used. I could quote no second example of equal swiftness, which yet in no way detracts from the elegance and free movements of the insect as it advances and retires, now on one side, now on another, without allowing the parabola of sand to stop. ”

“ The leg of an infinite branch may have at the extremity a tangent; this is an asymptote of the curve, and the leg is then hyperbolic; or the leg may tend to a fixed direction, but so that the tangent goes further and further off to infinity, and the leg is then parabolic; a branch may thus be hyperbolic or parabolic as to its two legs; or it may be hyperbolic as to one leg and parabolic as to the other. The epithets hyperbolic and parabolic are of course derived from the conic hyperbola and parabola respectively. ”

“ From the original form of this equation it appears that the two branches of this discriminant meet just inside the parabola in the end points with approximate co-ordinates (0.043, ±0.360) . The geometry of the cusp and end-points on the discriminant curve is suggestive of interesting development in detail. [Pg 63]
Values of λ and κ for points on the discriminant give curves with two modes coinciding. All points on one side of the discriminant have three real and distinct modes, and all on the other have one real and two imaginary modes. ”

“ It is a particular form of the limaçon (q.v.) and is generated in the same way. It may be regarded as an epicycloid in which the rolling and fixed circles are equal in diameter, as the inverse of a parabola for its focus, or as the caustic produced by the reflection at a spherical surface of rays emanating from a point on the circumference. ”

“ The segment between the asymptotes on any tangent to an hyperbola is bisected by the point of contact.
The first part allows a simple solution of the problem to find any number of points on an hyperbola, of which the asymptotes and one point are given. This is equivalent to three points and the tangents at two of them. This construction requires measurement.
§ 74. For the parabola, too, follow some metrical properties. A diameter PM (fig. 27) bisects every chord conjugate to it, and the pole P of such a chord BC lies on the diameter. But a diameter cuts the parabola once at infinity. ”

“ If I tilt up the plane and make the force of gravity big, the parabola is long and thin;88 if I weaken down the force of gravity by making the plane nearly horizontal, then it is wide and flat.
One can also show this by a stream of peas or shot. The little bullets go each with a uniform velocity horizontally, and an accelerated force downwards.
Instead of peas we can use water. A stream of it rushing horizontally out of an orifice will soon bend down into a parabola. ”

“ In the article Geometry, Analytical, it is shown that the general equation of the second degree represents a parabola Analytic Geometry. when the highest terms form a perfect square. This is the analytical expression of the projective property that the line at infinity is a tangent. The simplest equation to the parabola is that which is referred to its axis and the tangent at the vertex as the axes of co-ordinates, when it assumes the form where 2a=semilatus rectum ”

“ The metallic compounds formed by the combination of persulphomolybdic acid with a base are pulverulent, in many cases of a red colour, and for the most part insoluble in water. With barium, the acid furnishes a yellowish-red powder, insoluble in, but made denser by water, which imparts to it a cinnabar colour. With calcium it is said to yield a scarlet, sparingly soluble in water. With chromium, uranium, lead, platinum, and copper, it gives a dark red; that from the last metal turning brown when collected on a filter. It likewise produces reds with zinc, cadmium, iron, mercury, and tin ”

“ Porulosa or Holotrypasta.—The subclass Porulosa or Holotrypasta includes the two legions, Peripylea or Spumellaria, and Actipylea or Acantharia, which agree in the following constant and important characters:— (1) The Central Capsule is primitively a sphere, and retains this homaxon form in the majority of the species. (2) The Membrane of the central capsule is everywhere perforated by very numerous minute pores, but possesses no larger principal aperture (osculum) . (3) The Pseudopodia radiate in all directions and in great numbers from the central capsule, passing through its pores. ”

“ Hook-like and straight, jointed setæ are attached to the parapodia. Traces of a small enteric canal occur in the posterior half of the body.
Dimensions.—Length of specimen preserving 10 parapodia on the left side 33 mm. Width as flattened in the shale 3 to 4 mm. One parapodia is 3 mm. in length. Each one appears to be drawn in towards the body.
Observations.—The unique specimen representing this species is flattened and slightly distorted in the shale. ”

“ The body substance is granular, with a spherical, sharply contoured nucleus and an accessory nuclear body. No vacuoles are present, but occasionally the endoplasm contains red blood corpuscles. The pseudopodia are hyaline, finger- or lobe-shaped, and are protruded either singly or in twos. Multiplication is by binary fission and by the formation of spherical cysts (15 μ to 20 μ in diameter) in which occurs successive division of the nuclei, ultimately forming ten to twelve roundish bodies each of which soon develops a flagellum. ”

“ Instead of paraffining the pot in strips all round (as many operators do) paraffin the pot all round, except at one strip about half an inch wide, and let this face the carbon plate. If this be done, the difference in internal resistance between the cell with paraffined pot and the same cell with pot unparaffined will be little; but if the portion that is unparaffined be turned away from the carbon, it will make very nearly an additional 1 ohm resistance. It is necessary to have an ounce or so of mercury in each porous cell, covering the foot of the zinc ”

“ The trophozoite begins life as a small, rounded uninucleate corpuscle, which as it grows, becomes multinucleate. The multinuclear body generally assumes a definite shape, often that of a sausage. Later, the protoplasm becomes segregated around each of the nuclei, giving the parasite a mulberry-like aspect; hence this stage is frequently known as a morula. The uninuclear cellules thus formed are the spores, which are ultimately liberated by the break-up of the parent body. ”

“ For it may be observed, that the curvilinear paths ascribed to military projectiles by Rivius and Tartalea, and by other writers who followed them, as Digges and Norton in our own country, though utterly different from the theoretical form, the parabola, do, in fact, approach nearer the true paths of a cannon or musket ball than a parabola would do; and this approximation more especially exists in that which at first sight appears most absurd in the old theory; namely, the assertion that the ball, which ascends in a sloping direction, finally descends vertically. ”

“ Why the automatic mechanism controlling the direction of movement should produce a helical spiral in paramecium, a wavy path or flattened spiral in ameba, and a series of spirals in man, is not yet subject to profitable discussion, except of course to point out that paramecium is not restricted to two dimensions of space as is ameba and man. ”

“ Melt the paraffin mass over a water bath, take the specimen, and dry it between the folds of a cloth to remove the spirit, so that the paraffin may adhere to its surface, place it in a small chip-box, in the desired position, and pour in enough melted paraffin to cover it, then set aside to solidify; when quite cold break away the box, and cut sections from the embedded mass with a sharp razor. ”