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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122. If neither 0 nor r be considered the principal variable, it is required to prove that
{.+ (dr\
CP- = d.
/dr\2 drd20 d2r
^4-r- _ +r- - -- - r
+ \d0) do d02 d02
and PV= -.dd
2 dr\2 d r d20 d 2
r 2 Tl ( r — 1 d - -d ---
\d0] d do0 ddO
123. It is required to apply the criterion deduced in
Article (211) to ascertain whether the spiral whose equation
is r (e +e-0) =2a, is concave or convex towards its pole.
124. If the tangent to a spiral make an angle ( with
the rectangular ordinate, and p be the perpendicular let fall
upon it from the origin: it is required to prove that
ds d2p
+ dab =P + d2
where the upper or lower sign is to be used according as the
curve is convex or concave towards its pole.