TANGENT. In the article CONTACT we have given the first notion on this subject, which we now resume in a somewhat more general manlier, annexing the usual details of formulae, but without proof.
It is usual to apply the word tangent to the tangent straight line only, on which see DIRECTION generalising the definition, it will be RR follows :— Of all curves of a given species, or contained under one equation, that one (n) is the tangent to a given curve (A) at a given point, which passes through that given point, and is nearest to the curve (A) : meaning that no curve of the given species can pass through the given point, so as to pass between (B) and (A), immediately after leaving the point at which the two Latter intersect.
To ascertain the degree of contact of two curves which meet in a point, proceed as follows. Let y=tpx and y=tyx be the equations of tho curves, and a the abscissa at the point of contact; so that (pa = tya. At the point whose abscissa is a+ h, the difference of the ordinates of the curves is, by Taylor's theorem, as to which it will be found that A can be taken so small that the aeries shall be convergent, Now of two series of the form AA' + Rh. + + .... the value of that in which us is the greater will diminish with out limit as compared with the other, when A diminishes without limit. Consequently, every curve y=4.r, which has Va=tp'a, will approach, before the point of contact is attained, nearer to y tax than any other in which S'a is not =is'a. Again, when (lie x,co, those cams of yik.r in which ire= ea, will approach nearer to y ...or than any in which ea is not =ea ; and so on. Bence, to make y=1,tx have the closest possible contact with y=qxr when .r=a ;—give such values to the constants in as will satisfy as many as possible of the equations pa fa'a = ps'a = kc consecutively from the beginning. This is a brief sketch, which can be filled up from any elementary work ; and the following are the principal results :— 1. When the string of equations is satisfied up to the contact is said to be of the nth order.
2. In contact of the nth order, the deflection cp(a +A)-11,(a +A) diminishes with and vanishes in n finite ratio to it.
3. In contact of an even order, the curves intersect at the point of contact ; in contact of an odd order, they do not intersect at that point 4. When curves have a contact of the nth order, no curve, having with either a contact of an order inferior to the nth at tho same point, can pass between the two.
5. A straight line, generally speaking, can have only a contact of the , first order with a curve ; and the equation to the tangent straight line of the curve y=ox, when r= a, is y—fa= st/a(x —a). But if it should
happen that ea =0, =0, ke., up to = 0, then for that point the tangent has a contact of the nth order. Thus, at a point of con trary flexure the tangent has a contact of the second order, at least, with the curve.
6. A circle, generally speaking, can be made to have a contact of the second order with a curve, and the equation of the most tangent circle, or circle of Cenvasune, to the curve y=tpx, at the point x= a, is f itle (1 + tga 1 + ea ) + cp'a— a + • — — — + — cb as ys'a This circle cuts the curve, generally speaking: if not, as for example, at the vertices of an ellipso, it is evidence that the circle has a contact of some higher and odd order. The centre of the circle of curvature is a point on the normal, being that at which the normal touches the evolnte. [1strom= AND Evotrre.] Not only is the term tangent most generally applied to the closest straight line only, but frequently only to that portion of the straight line which falls between the point of contact and the axis of as Again, the normal is a straight line perpendicular to the buigent, drawn through the point of contact : but this term also is frequently applied only to that portion which falls between the point of contact and the axis of .r. It is with reference to this limitation that the terms sub tangent and subnormal are to be understood : the first meaning the distance from the foot of the tangent, to the foot of the ordinate ; the second that from the foot, of the ordinate to that of the normal. The formula for the subtangent is --epa-i-ea ; that for the subnormal (pa x dee. The sign deterimeing the mode of taking the line from the foot of the ordinate.
Let fl be the angle made by the tangent with the axis of x ; usually the angle made by that part of the tangent which has positive ordinates with the positive side of the axis of as Then /3, at the point whose abscissa is x, is determined by the equation If we take the more general mode of ine,asuretnent proposed in Sias, this equation remains equally true. Now, keeping strictly to that mode, let 13 be the angle made by the tangent with the axis of r, 0 the angle made by the radius vector r with the axis of a-, and a that made by the tangent with the radius vector. It will be found, then, that in all cases Unless the mode of attributing signs be carefully attended to, these last equations, though always considered as universally true, are not so in reality.