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

1345

POLAR FORMULX.

This result 191. This result we may also prove as follows.
deduced
from the By (175) ds V/dx2 + dy2',
rectangular
formula.
but ==rcosO, y = r sin 0;
dv = dr cosO. - r sin 0 d9,
dy= drsin90 + r cos 0 d8;
da + dy' di2 + r2 dO'; and.-. ds = Vdr2 r2dO2.
Proposi. 192. Draw SY perpendicular to the line QPT which
tion.
To find p passes through the points P and Q, and draw PO perpenthepepn
dicular on dicular to SQ, then by similar triangles
the tangent
from S. OP, rsin (0' - 0)
SY= SQ.
PQ c
sin (0'-O0) O'-0 s'-s
-Pr. '
0' - 0 ls
Now let 9' approach 0; then the limiting position of the
line QPT will be the tangent at P; and therefore if p be
the perpendicular from S upon that tangent, the limiting
value of SY will be p. Hence, by Lemmas IX and X,
we have
2 2
p r ds
Cor. 193. This result we may also prove as follows.
This result
deduced The equation to the tangent is
from rect.. h
angular
formule. dy
Y, - y ('Vj m),
d x
and the perpendicular p upon this line from the origin is
dy
dx ~~~xdy -ydx
dd, I OI~ dI —dor,since ds= Vdx' + di';
(d ~ds