An elementary treatise on the differential calculus, in which the method of limits is exclusively made use of, by the Rev. M. O'Brien.

CHAPTER XII.
TANGENTS AND NORMALS TO CURVES. THE CURVATURE OF CURVES.
THE EVOLUTE.
THE Differential Calculus is of great use in various parts of
analytical geometry. We have already seen in (26) that it puts
us in possession of a general method of drawing tangents to
curves. We now proceed to shew that it admits of, not only
this, but many other important applications in analytical
geometry.
posi- 173. If a right line SPQ (fig. 6) be drawn passing
To find the through any two points of a curve, and if SPT be its limiting
equation to
the tangent position when Q approaches P, i. e. if the angle made by PT
f a urent and PQ may be diminished ad libitum by sufficiently diminishing the arc QP; then PT is said to be a tangent to
the curve at the point P. Let x y be the co-ordinates of P,
'y' these of Q, Z PTX= f, then tan + is the limiting value
y --
of tan PRX when Q approaches P, i.e. of, Y when x'
approaches I, which, by definition of a differential coefficient,
dy
is -; hence
do
tan dy
tan ke = -.
Hence if x, y, be the co-ordinates of any point on the
tangent PT, since it is a line passing through the point P and
making an angle tan-1 dy with the axis of x, we have for its
dw
equation
dy
y,- y = - ('/ - _).