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

224 APPENDIX.
i 1
or /3 = 2rn + 7 a 31,
which is the equation to the evolute of a parabola.
It is represented by UWV (fig. 37).
(P.)
(1) Let the equation to the curve be
r = ael
then dr= 1ma e"' dO,
rdO
tan =- = - m.
dri
Hence in this curve, which is called the logarithmic spiral,
the tangent always makes the same angle with the radius
vector.
ds2 = 2 d02 + dr2 = a2e2 m (1 + m2) dO2;.'. ds - a / 1- + m. emOd = o,
/' + mu
d (. ' -- -- e' o) = 0;
%//1 + 2.. - a n --- —- = some constant, C suppose:
m
if s = 0 when 0 = 0, then.\/1 + m.- t
- a C;.. =, ----- ( - a?).
D/,
(2) The polar equation to an ellipse may be put in
the form
cos2 0 sin2 1
+ -...................(1);
a' b',r