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

THE EVOLUTE. 3

131

187. Differentiating the equations (1) we find Remarkable properties of
da = dx - -p cos +Idxk dp sin 'Jthis curve.
dj3= dy -p sin xfd4, + dpo cos 4P..
d s dxNow p Cos xkd+P = - T- d'q1 (183, 1-76),
dx,
and p sin xfrdxk dy. similarly;
hence dac = - dp sin4,
d/3 = dp cos~r
from which equations we find
dcc
da + d/32 d p2.......(3).
188. Let B'P'Q' (fig. 15) be the curve traced out by First Pro.
then the equation ~~~~~~~~~~~~perty.
P'; henthe quaionto the tangent to this curve at the The normal
point P' is PP'touches
the curve
d/f3 traced out
Yi - - x aby P' at
dac the point
or yi -/3=-cot (.QI1-a) by ()
Now this is evidently the equation to the line PP' since
that line passes through the point P", and makes an angle
4, + - with the axis of xv. Hence the normal PP' touches
the curve traced out by P" at the point P' as we have
represented in the figure.
189. Again, if a- denote the length of the arc B'P' of Second
Propry
the curve B'P'Q' measured from some fixed point B', we have Thoe curvtedo-=V'da 2~+dI32 = 4dp by(3): tae u
by ~~~~~~~by the endl
of a string
since p evidently decreases wheni o- increases in consequence unwound
off the
of the way we have measured a-, we must take the lower curve,of these signs., and therefore we have B'P'Q'o
da T+ dp = 0;