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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16. If two distances u, and u, from a fixed point be such
that u 2= -a2, it is required to compare their evanescent increments or decrements in any position.
17. Corresponding to the extremities of the latus rectum
of a common parabola, it is required to find the ratio of the
rates of increase of the abscissa and ordinate.
aa-n x- n
18. If u =,a it is required to shew that when
a-a
-=a, the value of u=aan( -- +loga}.
a J
sin mO0-cos(m.90-0)nO
19. Find the value of u= si- co.- when
sin mO+cos (n. 90~+m0)
0 =0, m being a whole number of the form 4t + 1 and n of
the form 4v + 3.
20. If 2y4 - Sy3 + X2y2_-aSx =0 it is required to find
the values of d when x = o and y= c.
dxv
21. In the equation x4y4- 2 aCX2y4 + ay2- a8 =0, it is
dy
required to find the values of - corresponding to =0 and
y= 00
22. If y = (a + bx) ex + m sin ax + n cos x, it is required
d4y d3y d2y dy
to prove that d -2 2 z-2- d+Y=O.
dx4 dX' dx' d
1 1 1.
23. If the equation of a curve be - -+- it is
y acn (n
required to prove that an x A T -= AMn+, where AM =,
MP=y and PT is a tangent.
24. Describe a parabola which shall touch a circle at
a given point, and have its axis coincident with a given diameter
of the circle.
25. If y and y' be the radii of curvature of a curve and
its evolute at corresponding points, it is required to prove that