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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196. Apply the result of the last example to the cases,
wherein the line of the proposed kind becomes a straight line
and a circle.
197. Let AEC, BFC and EF be three straight lines
given in position, and from the points A, B let straight lines
be drawn to any point in the straight line EF meeting AC in
P and BC in Q: then it is required to find the nature of
the curve to which the straight line joining P and Q is always
a tangent.
198. Prove that all the straight lines defined by the equation y - cx=-a/1 + c2 by assigning different values to c, are
tangents to the circle whose equation is y = /a -- 2.
199. Determine the nature of a curve when its tangent
is defined by the equation (c —l)x -cy=a - b, whatever be
the value of c.
200. Find the curve to which the straight line belonging
to the equation y = c (x - a) + b\/l + c2 is always a tangent,
whatever be the value of c.
201. Required the curve to which the straight lines
whose equations are y = c (a- x) +_ / 2 + c2a are always
tangents, whatever value be assigned to c.
202. If the equation to a straight line be - + -1
a /3
1 1 1.
where a and /3 are subject to the equation -- =, it
is required to prove that the curve formed by the intersections
of all such lines will be a circle whose equation is x2 + y2 = c2.
203. If an infinite number of straight lines defined by the
equation ay + 3x = a/3 be drawn, subject to the condition expressed by the equation a' + /3" = c', then will the curve whose
11 n n
equation is xn + + yn+1 = C + 1, touch them all.
204. Determine the nature of the curve which shall bound
all the parabolas expressed by the equation y = ax- (1 + a2) xV2,
whatever value be assigned to a.