683] 
ON THE FUNCTION arc SHI (x + iy). 
Suppose that X, p are the elliptic coordinates of the point (x, y) ; viz. 
have 
x 2 y 2 _ 
a 2 + X + b 2 + X~ ’ 
x 2 y 2 
a 2 + p + b 2 +p = ’ 
where a 2 + X, b 2 + X, and a 2 + p are positive, but b 2 + p is negative. Calling / 
distances of the point x, y from the points (c, 0) and (— c, 0), that is, assuming 
9 = - c) 2 + y 2 }, 
a = ^[(x + cf + y 2 ], 
then we have 
V(a 2 + X) = ^ (a + p), whence also \J(b 2 + X) = £ \/{{a + p) 2 — 4c 2 }, 
VO 2 + fi) = i (o' - p), » \/(b 2 + p) = \ */{{a ~ pf- 4c 2 }, 
which equations determine X, p as functions of x, y. 
Now we have 
pa = {(¿c 2 + y 2 — c 2 ) 2 — 4c 2 iu 2 } = \/ {{x 2 — y 2 — c 2 ) 2 + 4 x 2 y 2 }, 
p 2 + a 2 = 2 (x 2 + y 2 + c 2 ) ; 
substituting herein for x, y their values 
c sin |f cos ¿77, — ci cos |f sin 77?, 
we find 
whence 
Hence 
and 
a? — y 2 — c 2 = c 2 {sin 2 £ cos 2 ¿77 + cos 2 |f sin 2 ¿77 — (sin 2 |f + cos 2 |f) (sin 2 ¿77 + cos 2 ¿77)} 
= — c 2 (sin 2 £ sin 2 777 + cos 2 |f cos 2 177), 
(¿c 2 — y 2 — c 2 ) 2 = c 4 (cos 2 f COS 2 777 + sin 2 If sin 2 777) 2 
+ 4& 2 t/ 2 — 4c 4 sin 2 |f cos 2 |f sin 2 777 cos 2 777 
= c 4 (cos 2 If COS 2 777 — sin 2 If sin 2 777) 2 . 
2p<r = 2c 2 (cos 2 If cos 2 777 — sin 2 |f sin 2 777), 
p 2 + o- 2 = 2c 2 (sin 2 If cos 2 777 — cos 2 If sin 2 777 + 1) ; 
hence 
(p + a) 2 = 2c 2 (cos 2 777 — sin 2 777 + 1), = 4c 2 cos 2 777, 
(p — o-) 2 = 2c 2 (sin 2 |f — cos 2 1 +1), = 4c 2 sin 2 |f. 
Consequently 
a- + X = c 2 cos 2 777, and thence 6 2 + X = — c 2 sin 2 777, 
a 2 + p = c 2 sin 2 |f, „ b 2 + p = — c 2 cos 2 |f, 
values which verify as they should do the equations 
X 2 y 2 _ 
a 2 + X b 2 + X 
ft 2 , V 2 =1 
a 2 + p b 2 + p 
291 
that we 
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