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

POLAR FORMULAE.

137

This method of arriving at these formulse will enable us
to make out any of them very readily, in case we forget
them.
(For Examples of the use of these formulae see Appendix P.)

199. If + be the angle made by PT with
we have
+ = 0 +,, and..-.d = d0 + dp.

SA, then Proposition.
To find p
in terms of
r and p.

Now p = r sin r, and... dp = drsin(P + rcos(pd d
'rdO rdr
= dr -- + d d
ds ds
rdr rdr
=-T (do + d) ds
r dr. d r
_rd, since (183);
P ds p
rdr
hence p =- dp

200. We have as in (199)
rdO
= p + 0= tan-1 + 0,
dr)
hence differentiating, supposing dO constant, we have
rd2r
d, = d 0 d t
dr'2
+.1'k r

Proposition.
To find p in
terms ofr
and 0.

2dr2' - d2p + r2dO
= -- d --- — -- r, since ds' = dr2 + r2 d0
ds2 -