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

PREFACE.

v

ciple of Indeterminate Coefficients. Chapters X and
XI relate to Vanishing Fractions, and Maxima and
Minima, and contain some useful simplifications of the
common methods. The very insufficient and troublesome criterion usually employed in distinguishing the
maxima and minima of functions of two variables is
not introduced. Chap. XII relates to Tangents, Normals, &c., the Curvature of Curves, and the properties
of the Evolute; and here tht arrangement usually
adopted is somewhat departed from, and what seems
a more natural course pursued, in order to avoid certain difficulties, which I have observed very often impede the student on his first reading of the subject.
In Chap. XIII the useful Polar formulce and the
differentials of Areas, Volumes, &c., are deduced.
Chap. XIV relates to Asymptotes. Chap. XV contains a very simple method of tracing curves. Chap.
XVI relates to singular points. Chap. XVII contains the general Theory of Contacts and Ultimate
Intersections: no use is made of series in explaining
the different orders of contact. The remaining chapters are occupied with Elimination by differentiation,
Lagrange's Theorem, the properties of the Cycloid,
&c., &c. The Appendix contains Examples worked
out.
It was my intention to have added a few more
chapters, and among the rest, one on the origin and
progress of the Differential Calculus, and another on