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

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115. In any spiral, if 0 be the point of intersection of
perpendiculars to two consecutive radii vectores through their
extremities, it is required to prove that SP2 =SO.Sy.
116. In the focal distance SP of a parabola, SQ is taken
equal to the rectangular ordinate MP: it is required to draw
a tangent to the locus of the point Q.
117. If Q be a point so taken in the radius vector SP of
a parabola that SQ is equal to the perpendicular upon the
tangent from the focus: it is required to find the values of
tanP and p in the locus of Q.
118. Prove that the spiral whose equation is
(Or- 20) r=2a cosO,
cuts off equal arcs from all circles passing through its pole
and having their centres in its axis; and that the distances
from the pole at which the spiral cuts the axis are as - -
1 3 5
&c. the tangents of the angles in which it cuts it being as
1, 3, 5, &c.
119. Determine the asymptotic circle of the curve defined
by the equation 0 v/ar - = 1.
120. In any polar curve, if r be considered the independent variable, it is required to prove that the radius of
curvature is expressed by
p +
d20 dO /d\0)
r- +2- r2I + -
dr2 dr \drJ
121. If the arc s of the spiral be taken as the independent
variable, it is required to shew that
dO
ds
d )2 ds2
rdo ds2