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ON THE TANGENTS, NORMALS AND ASYMPTOTES TO CURVES.
Sect. 1. Rectilinear Co-ordinates.
If the equation to the curve be put under the form
y = f (x), the equation to a tangent at a point ry is
1– y = d.(2+ — x) ;
a' and y' being the current co-ordinates of the tangent. If the equation to the curve be put under the form
11 = Q(x, y) = C, the equation to the tangent takes the more symmetrical form
- (r' - x) + (y – y) = 0.
dx If u be a homogeneous function of n dimensions in x and y, by a well-known property of such functions
= nu = ne,
a da ty dy and the equation to the tangent becomes
d x ' dy The equations to the normal are
The perpendicular from the origin on the tangent is
por trdo GoT GOT
if u be a homogeneous function of n dimensions in a and y.
The portion of the tangent intercepted between the point of contact and the perpendicular on it from the origin is
- du du
ldu The portions of the axes cut off between the origin and the tangent, or the intercepts of the tangent, are
Hence the product of the intercepts of the tangent
= 1. Yo = 12Y = 4 m2 is constant ; and the triangle contained between the axes and the tangent, being proportional to this product, is also constant.
(2) The equation to the parabola referred to two tangents as axes is
Hence the equation to the tangent is
(a x) (by)
x 7 / 7
a b . l or X, Y. are the co-ordinates of the chord joining the points at which the ases touch the curve.
(5) The equation to one of the hypocycloids referred to rectangular co-ordinates is
21 + y; = a". The equation to the tangent is
Therefore 2, = aš x, y, = ay; and the portion of the tangent intercepted between the axes = (17" + y^).. = a; or the hypocycloid is constantly touched by a straight line of given
length which slides between two rectangular axes. The converse of this proposition, viz. that the locus of the ultimate intersections of a line of given length sliding between rectangular axes is this hypocycloid, was first shewn by John Bernoulli. (See his Works, Vol. 11. p. 447.)
For the perpendicular from the origin on the tangent we find
p = (a wy)}. (4) In the cissoid of Diocles,
The subtangent = a, and is therefore constant.
P = (a + y")' = (a + yo')}
(7) From the general parabolic equation
y" = an-lx, we find the equation to the tangent to be
m = (y - y) = 4 (2 – 3).
(8) In the curve x = €! we easily find, by taking the logarithmic differential,
Subtangent = ------- = - y - = - X, -.
X – Y
y (9) The equation to the cycloid referred to its vertex is
I dy (2a - x)
dx - x 7 AB (fig. 19) being the axis of a.
If M be the point where the ordinate meets the generating circle, and if we join ALA, MB, then
tan MAN =
v_MN_(20 x – 2”)}_dy
dv That is to say, the tangent to the cycloid is parallel to the chord of the generating circle. The normal is evidently parallel to the other chord MB. Hence also the angle which two tangents make with each other is equal to the angle between the corresponding chords of the generating circle.
Y. = – (2a X – X")! = PN - MN = PM. But from the generation of the curve, PM is equal to the arc of the circle AM, therefore yo = arc AM.