INFLECTION

, in Optics, called also Diffraction, and Deflection of the rays of light, is a property of them, by reason of which, when they come within a certain distance of any body, they will either be bent from it, or towards it; being a kind of imperfect reflection or refraction.

Some writers ascribe the first discovery of this property to Grimaldi, who first published an account of it, in his Treatise De Lumine, Coloribus, & Iride, printed in 1666. But Dr. Hook also claims the discovery of it, and communicated his observations on this subject to the Royal Society, in 1672. He shews that this property differs both from reflection and refraction; and that it seems to depend on the unequal density of the constituent parts of the ray, by which the light is dispersed from the place of condensation, and rarefied or gradually diverged into a quadrant; and this deflection, he says, is made towards the superficies of the opaque body perpendicularly.

Newton discovered, by experiments, this Inflection of the rays of light; which may be seen in his Optics.

M. De la Hire observed, that when we look at a candle, or any luminous body, with our eyes nearly shut, rays of light are extended from it, in several directions, to a considerable distance, like the tails of comets. The true cause of this phenomenon, which has exercised the sagacity of Des Cartes, Rohault, and others, seems to be, that the light passing among the eyelashes, in this situation of the eye, is inflected by its near approach to them, and therefore enters the eye in a great variety of directions. He also observes, that he found that the beams of the stars being observed, in a deep valley, to pass near the brow of a hill, are always more refracted than if there were no such hill, or the observation was made on the top of it; as if the rays of light were bent down into a curve, by passing near the surface of the mountain.

Point of Inflection, or of contrary flexure, in a curve, is the point or place in the curve where it begins to bend or turn a contrary way; or which separates the concave part from the convex part, and lying between the two; or where the curve changes from concave to convex, or from convex to concave, on the same side of the curve: such as the point E in the annexed figures; where the former of the two is con- cave towards the axis AD from A to E, and convex from E to F; but, on the contrary, the latter figure is convex from A to E, and concave from E to F.

There are various ways of finding the point of Inflexion; but the following, which is new, seems to be the simplest and easieft of all. From the nature of curvature it is evident that, while a curve is concave towards an axis, the fluxion of the ordinate decreases, or is in a decreasing ratio, with regard to the fluxion of the absciss; but, on the contrary, that the said fluxion increases, or is in an increafing ratio to the fluxion of the absciss, where the curve is convex towards the axis; and hence it follows that those two fluxions are in a constant ratio at the point of Inflection, where the curve is neither concave nor convex. That is, if x = AD the absciss, and y = DE the ordinate, then x^. is to y^. in a constant ratio, or x^./y^. or y^./x^. is a constant quantity. But constant quantities have no fluxion, or their fluxion is equal to nothing; so that in this case the fluxion of x^./y^. or of y^./x^. is equal to nothing. And hence we have this general rule: viz,

Put the given equation of the curve into fluxions; from which equation of the fluxions find either x^./y^. or y^./x^.; then take the fluxion of this ratio or fraction, and put it equal to 0 or nothing; and from this last equation find also the value of the same x^./y^. or y^./x^.: then put this latter value equal to the former, which will be an equation from whence, and the first given equation of the curve, x and y will be determined, being the absciss or ordinate answering to the point of Inflection in the curve.

Or, putting the fluxion of x^./y^. equal to 0, that is, , or , or , or , that is, the 2d fluxions have the same ratio as the 1st fluxions, which is a constant ratio; and therefore if x^. be constant, or x^.. = 0, then shall y^.. be = 0 also; which gives another rule, viz; Take both the 1st and 2d fluxions of the given equation of the curve, in which make both x^.. and y^.. = 0, and the resulting equations will determine the values of x and y, or absciss and ordinate answering to the point of Inflection.

For example, if it be required to find the point of Inflection in the curve whose equation is| . Now the fluxion of this is , which gives . Then the fluxion of this again made = 0, gives ; and this gives again . Lastly, this value of x^./y^. put = the former, gives ; and hence , or , and , the absciss.

Hence also, from the original equation, , the ordinate to the point of Inflection sought.

When the curve has but one point of Inflection, it will be determined by a simple equation, as above; but when there are several points of Inflection, by the curve bending several times from the one side to the other, the resulting equation will be of a degree corresponding to them, and its roots will determine the abscisses or ordinates to the same.

Other methods of determining the points of Inflection in curves, may be seen in most books on the doctrine of fluxions.

To know whether a curve be concave or convex towards any point assigned in the axis; find the value of y^.. at that point; then if this value be positive, the curve will be convex towards the axis, but if it be negative, it will be concave.

INFORMED Stars, or Informes Stellæ, are such stars as have not been reduced into any constellation; otherwise called Sporades.—There was a great number of this kind left by the ancient astronomers; but Hevelius and some others of the moderns have provided for the greater part of them, by making new constellations.

INGINEER. See Engineer.