\ 
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.