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

143

/d3y2 Id3X 2
and S-2=
(d2y d'3 d2X d 3y2'
dx2 dx3 dW2 dx3)
103. If the absolute centre of curvature be the point
where the intersection of two consecutive normal planes meets
the osculating plane, it is required to determine the coordinates of this point, and thence the absolute curvature.
104. If the arc s of the curve be considered the principal
variable, and a, J, 7 the.co-ordinates of the centre of absolute
curvature, it is required to shew that
d2x
ds2
/a-a= 2 2 /d 2 d 2
(ds2 + \ds2) + \d)
d2y
ds'
W _+ [_ +.
v (d )2 (d2y)2 (d2 \2'
d2z
ds2
7/d^^\2/~d2y\2 /Cry\d2
\ds2) + (da) + (d
and -_ 2
(d2 ()2y2 d2+ 2
idds] + \+ ds2 +
105. If the point under consideration be regarded as the
origin of co-ordinates and the tangent line as the axis of xa,
it is required to prove that
d2y d2z
a ds 3-28 d s2
2/ + {ds) [ds2) + \\d$