FOCUS

, in Geometry and the Conic Sections, is applied to certain points in the Ellipse, Hyperbola, and Parabola, where the rays reflected from all parts of these curves do concur or meet; that is, rays issuing from a luminous point in the one focus, and falling on all points of the curves, are reflected into the other Focus, or into the line directed to the other Focus, viz, into the other Focus in the ellipse and parabola, and directly from it in the hyperbola. Which is the reason of the name Focus, or Burning-point. Hence, as the one Focus of the parabola is at an infinite distance; and consequently all rays drawn from it, to any finite part of the curve about the vertex, are parallel to one another; therefore if rays from the sun, or any other object so distant as that those rays may be accounted parallel, fall upon the curve of a parabola or concave surface of a paraboloidal figure, those rays will all be reflected into its Focus.

Thus, the rays Pf, from the Focus f, are reflected in the direction PF, into the other Focus F, in the ellipse and parabola, and form the Focus F, into FQ, in the hyperbola.

In all the three curves, the double ordinate CD drawn through the Focus F, is the parameter of the axis, or a 3d proportional to AB and ab, the transverse and conjugate axes.

In the ellipse and parabola, the transverse axis is equal to the sum of the two lines PF + Pf, drawn from the Foci to any point P in the curve; but in the hyperbola, the transverse is equal to the difference of those two lines. That is, in the ellipse and parabola, in the hyperbola.

In the ellipse and parabola, the square of the distance between the Foci, is equal to the difference of the squares of the two axes; and in the hyperbola, it is equal to the sum of their squares: that is in the ellipse and parabola. in the hyperbola.

Therefore the two semi-axes, with the distance of the Focus from the centre, form always a right-angled triangle FaE, or AaE.

In all the curves, the conjugate semi-axis is a mean proportional between the distances of either Focus from either end of the transverse axis: that is, AF : Ea :: Ea : FB, or Ea²=AF . FB.

If there be any tangent to these curves, and two lines drawn from the Foci to the point of contact; these two lines will make equal angles with that tangent. So, if GPG touch the curve at P; then is the angle FPG = [angle]fPG.

If a line be drawn from either Focus, perpendicularly upon a tangent; the distance of their intersection from the centre will be equal to the semi-transverse axis. So, if FH or fH be perpendicular to the tangent PH; then is EH=EA or EB. Consequently, the circle described on the diameter AB, will pass through all the points H.

The foregoing are the chief properties that are common to the Foci of all the three conic sections. To which may be added the following properties which are peculiar to the parabola: viz,

In the parabola, the distance from the Focus to the vertex, is equal to 1/4 of the parameter, or half the ordinate at the Focus: viz, AF=1/2 FC.

Also, a line drawn from the Focus to any point in the curve, is equal to the sum of the Focal distance from the vertex and the absciss of the ordinate to that point: i. e.

If from any point of a parabola there be drawn a tangent, and a perpendicular to it PK, both to meet the axis produced; then the focus will be equally distant from the two intersections and the point of contact: i. e. FG=FP=FK. |

Hence also the subnormal IK is=2AF or=FC the semi-parameter.

The line drawn from the Focus to any point of the curve, is equal to 1/4 the parameter of the diameter to that point: i. e. FP=1/4 the parameter of the diameter Pf.

If an ordinate to any diameter pass through the Focus, it will be equal to half its parameter; and its absciss equal to 1/4 of the same parameter; or the absciss equal to half the ordinate: i. e. PL=(1/4)MN=(1/2)LM or (1/2)LN.

, in Optics, is a point in which several rays meet, and are collected, after being either reflected or refracted. It is so called, because the rays being here brought together and united, their force and effect are increased, insomuch as to be able to burn; and therefore it is that bodies are placed in this point to be burnt, or to shew the effect of burning glasses, or mirrors.—It is to be observed however, that in practice, the Focus is not an absolute point, but a space of some small breadth, over which the rays are scattered; owing to the different nature and refrangibility of the rays of light, and to the imperfections in the figure of the lens, &c. However, the smaller this space is, the better, or the nearer to perfection the machine approaches. Huygens shews that the Focus of a lens convex on both sides, has its breadth equal to 5/8 of the thickness of the lens.

Virtual Focus, or Point of Divergence, so called by Mr. Molynenx, is the point from whence rays tend, after refraction or reflection; being in this respect opposed to the ordinary Focus, or Point of Concurrence, where rays are made to meet after refraction or reflection. Thus, the Foci of an hyperbola are mutually Virtual Foci to each other; but, in an ellipse, they are common Foci to each other: for the rays are reflected from the other Focus in the hyperbola, but towards it in the ellipse; as appears by the figures at the beginning of this article.

And, in Dioptrics, let ABC be the concavity of a glass, whose centre is D, and axis DE: Let FG be a ray of light falling on the glass, parallel to the axis DE; this ray FG, after it has passed through the glass, at its emersion at G will not proceed directly to H, but be refracted from the perpendicular DG, and will become the ray GK, which being produced to meet the axis in E, this point E is the Virtual Focus, as the ray is refracted directly from this point.

1. The Focus of a convex glass, i. e. the point where parallel rays transmitted through a convex glass, whose surface is the segment of a sphere, do unite, is distant from the pole or vertex of the glass, almost a diameter and half of the convexity.—2. In a PlanoConvex glass, the Focus of parallel rays is distant from the pole of the glass a diameter of the convexity, if the segment do not exceed 30 degrees. Or the rule in Plano-Convex glasses is, As 107 : 193 :: so is the radius of convexity: to the refracted ray taken to ita concourse with the axis; which in glasses of larger spheres is almost equal to the distance of the Focus taken in the axis.—3. In Double Convex glasses of the same sphere, the Focus is distant from the pole of the glass about the radius of the convexity, if the segment be but 30 degrees. But when the two convexities are unequal, or segments of different spheres, then the rule is, As the sum of the radii of both convexities: to the radius of either convexity alone :: so is double the radius of the other convexity: to the distance of the Focus. —Here observe, that the rays which fall nearer the axis of any glass, are not united with it so near the pole of the glass as those farther off: nor will the Focal distance be so great in a plano-convex glass, when the convex side is towards the object, as when the plane side is towards it. And hence it is truly concluded, that, in viewing any object by a plano-convex glass, the convex side should always be turned outward; as also in burning by such a glass.

1. That in Concave glasses, when a ray falls from air parallel to the axis, the Virtual Focus, by its first refraction, becomes at the distance of a diameter and a half of the concavity.—2. In Plano-Concave glasses, when the rays fall parallel to the axis, the Virtual Focus is distant from the glass, the diameter of the concavity.—3. In Plano-Concave glasses, as 107 : 193 :: so is the radius of the concavity: to the distance of the Virtual Focus.—4. In Double Concaves of the same sphere, the Virtual Focus of parallel rays is at the distance of the radius of the concavity. But, whether the concavities be equal or unequal, the Virtual Focus, or point of divergency of the parallel rays, is determined by this rule; As the sum of the radii of both concavities: is to the radius of either concavity :: so is double the radius of the other concavity : to the distance of the Virtual Focus.—5. In Concave glasses, exposed to converging rays, if the point to which the incident ray converges, be farther distant from the glass than the Virtual Focus of parallel rays, the rule for finding the Virtual Focus of this ray, is this; As the difference between the distanoe of this point from the glass, and the distance of the Virtual Focus from the glass: is to the distance of the Virtual Focus :: so is the distance of this point of convergence from the glass: to the distance of the Virtual Focus of this converging ray.—6. In Concave glasses, if the point to which the incident ray converges, be nearer to the glass than the Virtual Focus of parallel rays, the rule to find where it crosses the axis, is this; As the excess of the Virtual Focus, more than this point of-convergency: is to the Virtual Focus :: so is the distance of this point of convergency from the glass: to the distance of the point where this ray crosses the axis. |

1. To find, by experiment, the Focus of a convex spherical glass, being of a small sphere; apply it to the end of a scale of inches and decimal parts, and expose it before the sun; upon the scale may be seen the bright intersection of the rays measured out: or, expose it in the hole of a dark chamber; and where a white paper receives the distinct representation of distant objects, there is the Focus of the glass.—2. For a glass of a pretty long Focus, observe some distant object through it, and recede from the glass till the eye perceives all in confusion, or the object begins to appear inverted; then the eye is in the Focus.—3. For a Plano-Convex glass: make it reflect the sun against the wall; on the wall will then be seen two sorts of light, a brighter within another more obscure: withdraw the glass from the wall, till the bright image be in its least dimensions; then is the glass distant from the wall about a fourth part of its Focal length.—4. For a Double Convex: expose each side to the sun in like manner; and observe both the distances of the glass from the wall : then is the first distance about half the radius of the convexity turned from the sun; and the second is about half the radius of the other convexity. The radii of the two convexities being thus known, the Focus is then found by this rule; As the sum of the radii of both convexities : is to the radius of either convexity :: so is double the radius of the other convexity : to the distance of the Focus.

Dr. Halley has given a general method for finding the Foci of spherical glasses of all kinds, both concave and convex; exposed to any kind of rays, either parallel, converging, or diverging; as follows: To find the Focus of any parcel of rays diverging from, or converging to, a given point in the axis of a spherical lens, and making the same angle with it; the ratio of the sines of refraction being given.

Suppose GL a lens; P a point in its surface; V its pole; C the centre of the spherical segment; O the object, or point in the axis, to or from which the rays proceed; and OP a given ray : and suppose the ratio of refraction to be as r to s. Then making CR to CO, as s to r for the immersion of a ray, or as r to s for the emersion (i. e. as the sines of the angles in the medium which the ray enters, to the corresponding sines in the medium out of which it comes); and laying CR from C towards O, the point R will be the same for all the rays of the point O. Lastly, drawing the radius PC, continued if necessary; with the centre R, and distance OP, describe an are intersecting PC in Q. The line QR, being drawn, shall be parallel to the reflected ray; and PF, being made parallel to it, shall intersect the axis in the point F, the Focus sought. —Or, make as CQ : CP :: CR : CF, which will be the distance of the Focus from the centre of the sphere. —And from this general construction, he adverts to a number of particular simple cases.

Dr. Halley gave also an universal algebraical theorem to find the Focus of all sorts of optic glasses, or lenses. See the Philos. Trans. N° 205, or Abr. vol. 1, pa. 191.

These are easily found for any known curve, from this principle, that the angle of reflection is always equal to the angle of incidence.

The same are also easily found by experiment, being exposed to any object.

The increase of heat from collecting the sun's rays into a Focus, has been found in many cases of burning glasses, to be astonishingly great; the effect being increased as the square of the diameter of the glass exceeds that of the Focus. If, for instance, there be a burning glass of 12 inches diameter; this will collect or crowd together all the rays of the sun which fall upon the glass into the compass of about 1/8 part of an inch : then, the areas of the two spaces being as the square of 12 to the square of 1/8, or as the square of 96 to the square of 1, that is, as 9216 to 1; it follows, that the heat in the Focus will be 9216 times greater than the sun's common heat. And this will have an effect as great as the direct rays of the sun would have on a body placed at the 96th part of the earth's distance from the sun; or the same as on a planet that should move round the sun at but a very little more than a diameter of the sun's distance from him, or that would never appear farther from him than about 36 minutes.

Besides Dr. Halley in his method for finding the Foci, several other authors have written upon this subject; as Mr. Ditton, in his Fluxions; Dr. Gregory, in his Elements of Dioptrics; M. Carré, and Guisnée, in the Memoires de l'Acad.: Dr. Barrow and Sir I. Newton have also neat and elegant ways of finding geometrically the Foci of spherical glasses; which may be seen in Barrow's Optical Lectures.