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

CHAP. IX.
MISCELLANEOUS EXAMPLES.
BY means of the principles laid down in Articles (190)(212), it is required to prove and solve the following Theorems
and Problems.
1. If 0 and r be the co-ordinates of the logarithmic spiral
corresponding to the arc s, it is required to shew that
d s/ 1 q+ (log a)= aOV/1 + (log a) =r/ + (log a)2; d- log a)
dO dr log a
2. In the lemniscata of Bernoulli whose equation is
r2 a cos20, it is required to prove that
ds a a2 ds 1 a2
d - -; and
d0 ~ /cos 20 r dr sin 20 /ai3. In the spiral of Archimedes whose equation is r = aO,
it is required to prove that the tangent of the angle between
the radius vector and a straight line through two points whose
(2 a —/) sin (a-/3)
abscissae are a, 3 is expressed by (2a- - )os (a-a), the
(2 a - /3) cos (a-3) - a
said radius vector belonging to the abscissa a.
4. In every curve referred to polar co-ordinates, it is
required to establish the formulae
rd0 dr dO
sin P= -- cos P= - and p=r2.
ds ds ds
5. The equation of the reciprocal spiral is r = a0-1; then,
if the notation of Article (193) be retained, tan P=- - and
or
ar
p= - -: required a proof.
Iat& + r2