103. Let x be the value of x' in the equation to the tangent at any point of a curve when y' = 0. The length x, viz. OT' in fig. (1), is called the intercept of the tangent on the axis of x. It is easily seen from the two forms of the equation to the tangent that du du x + y dx dy du dx in the same way, y, denoting the intercept of the tangent on the 104. The portion MT, fig. (1), of the axis of x, contained between its intersections with the ordinate and the tangent at P, is ordinarily called the subtangent at the point P. It is evident, from the equations to the tangent, that MT is equal to Length of the Tangent. 105. The word tangent is sometimes used to denote the finite line PT, fig. (1), included between the point P of contact of the indefinite tangent HK with the curve and the point T in which the indefinite tangent cuts the axis of x. In this sense of the word it is plain that the length of the tangent is equal to 106. The finite lines PG, GM, fig. (1), G being the intersection of the indefinite normal UV and the axis of x, are frequently called the normal and subnormal respectively. It is plain that the subnormal is equal to the value of x' - x deduced from either of the equations to the normal, when y' = 0; or Form of the Equation to the Tangent to Curves of which the equations involve only rational functions of x and y. 107. Let u=0 be the equation to a curve; and suppose that u, denoting a homogeneous function of x and y ofr dimensions. Then the equation to the tangent at any point x, y, will be LEMMA. Let v be any rational function of x and y, = = 0. of where c is a constant coefficient, and a + ß=r, the term cxa y being a type of all the terms. Then Ex. To find the equation to the tangent at any point of the conic section ax2 + by3 + 2cxy + 2a'x + 2b'y + c' By the above formula we see at once that x' . (x' + cy + ax) + y' . (b' + cx + by) + c' + d'x + b'y = 0. Oblique Axes. 108. The forms of the equations to the tangent are not altered if we suppose the axes to be oblique instead of rectangular. Let P, P,, (fig. 3) be the two points in which the secant H'K' cuts the curve AB, referred to oblique axes Ox, Oy. Draw PM, P,M1, P'M', parallel to yO and cutting Ox in M, M, M', P' being any point whatever in the secant H'K'. Draw Ppp', parallel to Ox, and cutting P,M,, P'M', in p1, p'. Let OM=x, PM=y, OM1=x1, P ̧M1=y1, OM' =x', P'M'=y'. Then, by the similarity of the triangles P'p'P, P‚μ‚P, we have In the limit, when P, moves indefinitely near to P, x, y1 become x, y, respectively, and |
