\
292
on the function are sin (x + iy).
[683
viz. these become
C 2 COS 2 %r)
+
t
C 2 Sir 17]
— = sin 2 £ + cos 2
= 1,
x 1 y 2
c 2 sin 2 f ^ — c 2 cos 2 £
COS 2 ¿7/ + sin 2 it), — 1.
The same equations, or as we may also write them,
X = — a 2 sin 2 iy — 6 2 cos 2 it],
= — a? cos 2 £ — 6 2 sin 2 £,
determine 77 as a function of X, and £ as a function of /x; X, ¡x being by what
precedes, given functions of x, y.
Or more simply, starting from the last-mentioned values of X, /a, and substituting
these in the expressions
we find
or say
whence
x‘ =
a 2 + X . a 2 + /7
a 2 - b 2
, _ & 2 + X . b 2 + /1
y ~ *
X 2 = C 2 sin 2 £ COS 2 17), y 2 = — C 2 COS 2 £ sin 2 17),
x — c sin £ cos 777, iy = c cos £ sin 777,
x+iy = c sin (£ + 777),
the original relation between x, y and f, 77.