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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and 2 f=
d2y 2 \ d 2 2
ds2) + ds2
106. If the equation of a curve surface be of the form
/ du\2 (`du\2 /du\ 2
u =f(x, y, ) = 0, and 2= (- + - + (, and
VlaJ v\dyI kd v7
the notation of Article (297) be retained, it is required to
I du 1 du 1 du
prove that cos a =- - cos =- - and cos y=- -
d x JL dy / u dz
107. Determine the equation to the curve traced out
upon any of the co-ordinate planes by the intersection of
the tangent line to a curve of double curvature whose equations are y2=2a — 2 and 2 = m (x y2).
108. Prove that the differential of the area of the surface
traced out by the ordinate z in any curve of double curvature
-= z/da2 + dy2.
109. Trace the surface whose equation is x 2 + y2 = a2,
and determine the positions of the tangent planes at the points
which it cuts the co-ordinate axes.
110. Find the form of the surface defined by the equation - a + (Y — + (z- ) = 1, determining the
a b C
natures of its principal sections.
111. If an ellipsoid be cut by a plane passing through
the origin perpendicular to the plane of xy, it is required to
prove that the normals to the surface drawn from every point
of the intersection of the plane with the ellipsoid will cut
the plane of xy in a straight line, and to find the equation
to that line.
112. If the partial differential equation of a curve surface be z = a + px + qy, it is required to shew that there is
some point from which all the perpendiculars, let fall upon
the tangent plane at every point, are of an invariable length.