FOG
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FOG
("cribed: Or, two Points in the longer Axis, whence two Right-lines being drawn to any Point in the Circumference, jhall be together equal to the Axis it felf. Thefe are alio called Umbilici. See Umbilicus, and Centre.
To find the Foci of an Ellipfis, from B to L, fet off half the Parameter : And in the Centre C ereS a Perpendicular C K, meeting a Semicircle defcribed on A L. Then making C F = C K ; the Point F will be the Focus.
If then the Axis A B be cut in the Fccus F; the Rea~ angle of the Segments of the Axis A F, F B, 'will be fub- quadruple of the Reflangle of the Parameter into the Axis; whence the Square of the Diftance of the Focus from the Centre is equal to the Rcftangle of half the Axis into the Difference of the Semi-parameter from half the Axis. See Ellipsis.
Focus of the Hypericin, fee Hyperbola.
Focus, in Opticks, is a Point wherein fcveral Rays con- cur, and are colkaed ; either after having undergone Re- fraction, or Reflection. See Ray.
It is thus called, by reafon the Ravs being here brought together, and united = their Force and'Effea is increas'd ; fo that they become able to burn : Accordingly, 'tis in this Point, that Bodies are plac'd to fuftain the Force of Burning Glaffes, or Mirrors. See Burning Glafs, &c.
It muft be obferv'd, that the Focus is not flriftly fpeak- ing a Point ; the Rays are not all accurately collected into the fame Place : Huygens demonstrates, that the Focus of a Lens convex on both Sides is |- of the Thicknefs of the Lens. See Lens. — —
Focus, in Dioptricks, is the Point wherein the refrafled Rays, by Refrafiion render'd convergent, do concur, or meet, and crofs the Axis. See Refraction.
The fame Point is alfo call'd the 'Point of Concourfe, or Concurrence. See Point of Concourfe.
Virtual Focus, in Dioptricks, is the Point from which refrafted Rays, when by Refraftion they are render'd di- vergent, do begin to diverge, or recede from each other. See Virtual Focus.
The fame Point is alfo call'd Puntfum Diffcrfus. See Point of 5Difperflon.
The Effeft of convex Glaffes, or Lenfes, is to render the Rays, tranfmitted thro' them, convergent, and bring them together into a Focus, which will be nearer or further off, as the Lens is a Portion of a greater or lefs Sphere. See Convex, and Convergent.
The Effect of concave Lenfes is to render the Rays tranf- mitted thro' them, divergent, or to difperfe them from a Virtual Focus. See Concave, and Divergent.
For the 'Place, Pafuion, Diftance, Sec. of the Foci of Rays refraSed thro' 'Plain, 'Concave, and Convex Medi- ums of divers Denjities, as Air, Water, Glafs, &c. fee Re-
FRACTION, £i?C.
The Laws of the Foci of Glaffes, and the Methods of finding the fame, being thofe of moil Ufe, and Importance; we fhall here fubjoin them a-part, as deliver'd and demon- ftrated by the ingenious Mr. Atolyncux, in his 'Dioftrica Nova.
i° then, The Focus of a convex Glafs, i. c. the Point wherein parallel Rays tranfmitted through a convex Glafs, whole Surface is the Segment of a Sphere do unite, is di- ftant from the Pole, or Vertex of the Glafs, almoll a Diameter and half of the Convexity.
z° In a Piano- Convex Glafs, the Focus of parallel Rays, or the Place where they unite with the Axis, is diftant from the Poie of the Glafs a Diameter of the Convexity; pro- vided the Segment do not exceed 30 Degrees.
The Rule or Canon in Piano-Convex Glaffes, is as 107 : 193 : : So is the Radius of the Convexity : to the refracled Ray taken in its Concourfe with the Axis; which in Glaffes of larger Spheres, is almoll equal to the Diftance of the Fo- cus taken in the Axis.
3° In double Convex Glaffes of the fame Sphere, the Fo- cus is diftant from the Pole of the Glafs about the Radius of the Convexity, if the Segment be but 30 Degrees.
But it the Convexities arc uneqyal, or if the two Sides arc 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 the double Ra- dius of the other Convexity : to the Diftance of the Focus.
Here obfervc, that the Rays which fall nearer the Axis of any Glafs, are not united with it fo foon as thofe farther off: Nor will the Focal 'Diftance be fo great in a Plano- Convcx Glafs, when the convex fide is towards the Object, as when the contrary way.
Hence it is truly concluded, that in viewing any object, by a Piano Convex Glafs, the convex Side ftiould be turned out- ward ;, as alfo in burning by iuch a Glafs.
For the Virtual Focus obferve,
i° That in Concave Glaffes, when a Ray falls from Air parallel to the Axis, the Virtual Focus, by its firft Refraction becomes at the diftance of a Diameter and an half of the Concavity.
i g In 'Piano-Concave Glaffes, when the Rays fall parallel
to the Axis, the Virtual Fccus is diftant from the Glafs thg Diameter of the Concavity.
3° InPlauo-ConcaveGlaf/es; as 1C7 : 193 :: So is the Radius of the Concavity, to the Diftance of the Virtual Focus.
4° In double Concaves of the fame Sphere, the Virtual Focus of parallel Rays is at the Diftance of the Radius of the Concavity.
But whether the Concavities be equal or unequal, the Vir- tual 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 the double Radius of the other Concavity : to the diftance of the Virtual Focus.
5° In Concave Glaffes, if the Point to which the inciden. Ray converges, be diftant from the Glafs farther than the Virtual Focus of parallel Rays ; the Rule for finding the Virtual Focus of this Ray is this ;
As the Difference between the Diftance of this Point froiri the Glafs, and the Diftance of the Virtual Focus from the Glafs : is to the Diftance of the Virtual Focus : : So the Di- ftance of this Point of Convergence from the Glafs : is to the Diftance ot the Virtual Focus of this converging Ray.
6° In Concave Glaffes, if the Point to which the incident Ray converges, be nearer to the Glafs, than the Virtual Fo- cus of parallel Rays ; the Rule to find where it croffes the Axis, is this ;
As the Excefs of the Virtual Focus, more than this Point of Convergency : is to the Virtual Focus : : So the Diftance of this Point of Convergency from the Glafs : is to the Di- ftance of the Point where this Ray croffes the Axis.
Practical Rules fir finding the Foci of Glaffes.
To find the Focus of a Convex Spherick Glafs, being of a fmall Sphere, apply it to the End of a Scale of In- ches, and Decimal Parts, and expofe it before the Sun ; upon the Scale you will have the bright Interfection of the Rays meafur'd out ; or, expofe it in the Hole of a dark Chamber ; and where a white Paper receives the diftinct Reprefentation of diftinfl Objects, there is the Focus of the Glafs.
For a Glafs of a pretty long Focus ; obferve fome dif- tant Objefts thro' it, and recede from the Glafs till the Eye perceives all in Confufion, or the Objefls begin to appear inverted ; here the Eye is in the Focus.
For a Piano-Convex Glafs: Make it reflect, the Sun i- gainft a Wall ; you will on the Wall perceive two Sorts of Light ; one more bright within another more obfeure : Withdraw the Glaffes from the Wall, till the bright I- mage is at its fmalleft ; the Glafs is then diftant from the Wall about the fourth Part of its focal Length.
For a double Convex : Expofe each Side to the Sun in like manner ; and obferve both the Diftances of the Glafs from the Wall. The firft Diftance is about half the Radius of the Convexity turned frrm the Sun ; and the fecond, about half the Radius of the other Convexity.
Thus, we have the Radii of the two Convexities; whence the Focus is found by this Rule ;
As the Sum of the Radii of both Convexities : is to the Radius of either Convexity : : So is the double Radius of the other Convexity : to the Diftance of the Focus.
Focus, in Catoptricks, is a Point wherein the Rays re- flexed from the Surface of a Mirror, or Speculum ; and by Reflection render'd convergent; do concur, or meet. See Mirror.
The Effecl of Concave Mirrors is to colleft the Rays fal- ling on their Concave Surface into a Focus. See Concave Mirror.
The Effeft of Convex Mirrors, is to difperfe the Rays falling on them, or render them more Divergent. See Con- vex Mirror.
For the Laws of the Foci of Rays refraSed from Mir- rors, or Specula, fee Reflection, &c.
The Foci of Concave Glaffes are had by Refkaion : For as a Concave Mirror burns at the Diftance of about half the Radius of the Concavity; fo a Concave Glafs, being fup- pos'd a reflefling Speculum, unites the Rays of the Sun, at the Diftance of about half the Radius of the Concavity. To find the Foci of all Glaffes, Geometrically ;
Dr. Halley furnifhes us a general Method for finding the Foci of fpherical Glaffes of all kinds, both concave, and convex ; expos'd to any kind of Rays, either parallel, con- verging, or diverging ; under the following Problem.
To find the Focus of any Parcel of Rays diverging from, or converging to a given Point in the Axis of a fpherical Lens, and inclined thereto under the lame Angle: The Ra- tio of the Sines of Refraaion being given :
Suppofe GL {Tab. Opticks, Fig. 38.) a Lens; P a Point in its Surface ; V its Pole; C the Centre of the Sphere whereof it is a Segment; O, the Objea, or Point in the Axis, to or from which tho Rays do proceed ;and O P a given Ray : And fuppofe the Ratio of Refraftion to be as r to s.
Then,
