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

188

POLAR FORMULIE.

hence - d s =d
d+11 dO (2dr' - rd2r + r'd8')
ds'
or dO (dr' + dS' - rd~r)
Cor. 201. This result may also be arrived at thus.
The same
result de. ds3
from We have by (186) =dydx d'd
formuhe.
Now dy = drsin 8 + rcoA 18,
d2y = d r2sin8 + 2 drcos8d8 - rsin8d 8,
dx = drcos 8 - r sin 8 d 8.
Hence
d'ydx = (drd'r - Srdrd 82) sin 8cos 8
+ 2 dr 2cos28. d8 0- r (d'r - rd 80) sin28. do.
Now d'xdy is evidently got from this by putting ~ - 8
2
instead of 8; hence we evidently, have, putting sin 8 for cos 8,
cos 8 for sin 8, - d8 for d8, and subtracting
d'ydx - d'xdy = dO (2 dr2 - rd2r + r'd8'),
ds'
and.~. p~~~~~d.3
and'.. P- d (2dr' - rd'r + r 2d8)
Proposi. 2021 If p increases when r increases, it is easy to see that
Tfind the concavity of the curve is turned towards the pole, and if
whether the p diminishes when r increases, it is turned from the pole.
concavity
ofthecurve Hence, by Lemma XVIII., the concavity of the curve is
is turned to
or from the dp
pole. turned towards or from the pole according as T is positive
or negative.
(For Examples see Appendix P.)
The use of the following formulae respecting the differentials of areas, surfaces, and volumes cannot be shrewn
without the assistance of the Integral Calculus.