The elements of non-Euclidean plane geometry and trigonometry, by H. S. Carslaw.

71, 72]

POLAR COORDINATES

123

The figure PRQS becomes a rectangle in the limit, and we
can use the Euclidean expression for its area (cf. ~ 70).

Then

arc PR = cosh & x [~69 (2)]
le [6()

and PS = 8y.
Hence the element of area in Cartesian Coordinates is
coshY dx dy.
k
~ 72. The Element of Area in Polar Coordinates.
As before, the result can be obtained by using the equations
cosh - = cosh - cosh,
k k k
tanh Y
tan 0 = —,
sinh
k
which connect (r, 0) and (x, y).
Q

R

p

0
FIG. 84.
But it is simpler to obtain the element of
from the geometrical figure:
Let P, Q be the points (r, 0), (r + 8r, 0 + 60).

area directly