Distance of the Origin of Coordinates from the Tangent.

102. Let & denote the distance of the origin from the tangent, being the inclination of 8 to the axis of x. Then the equation to the tangent must coincide with the equation

x' cos ε + y' sin ɛ = S.

Comparing this equation with each of the equations to the tangent, viz.

103. Let x be the value of x' in the equation to the tangent Хо 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 of r 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, of

where c is a constant coefficient, and a + ß=r, the term cx y being a type of all the terms. Then

du du +y

=

dx dy

(➖ ➖ nu − (n − 1) u, − (n − 2) u, − (n − 3) U2- ... -2-2--1

and therefore, from the equation

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

= 0.

x'. (a+cy + ax) + y'. (b' + cx + by) + c' + dx + by = 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, P1, (fig. 3.) be the two points in which the secant H'K' cuts the curve AB, referred to oblique axes Ox, Oy.

Draw PM, PM1, P'M', parallel to y0 and cutting Ox in M, M1, M', P' being any point whatever in the secant H'K'. Draw Pp,p', parallel to Ox, and cutting P,M,, P'M', in p1, p. Pv Let OM=x, PM=y, OM1=x1, P1M1=y1, OM' = x', P'M' =y'. Then, by the similarity of the triangles P'p'P, P ̧ð ̧P, we have

In the limit, when P, moves indefinitely near to P, x1, Y, become x, y, respectively, and