[959 
960] 
537 
8, 
K. 
their 
sh we 
960. 
ON THE CIECLE OF CUEVATUEE AT ANY POINT OF 
AN ELLIPSE. 
[From the Messenger of Mathematics, vol. xxiv. (1895), pp. 47, 48.] 
is 
Let 
u 2 + v 2 = 1. 
The equation of the circle of curvature at the point (x, y) of the ellipse 
X 2 + Y 2 + 2 —(— bXv? + a Yv 3 ) + u 2 (a 2 — 2b 2 ) + v 2 (b 2 — 2a 2 ) = 0. 
Write X = a£, Y = by, then this becomes 
x i y* 
a 2 b 2 
1 
a 2 | 2 + h 2 y 2 + 2 (a 2 - 6 2 ) (- %u s + yv 3 ) + u 2 (a 2 - 2b 2 ) + v 2 (b 2 - 2a 2 ) = 0. 
To find where this meets the ellipse, we must write £ 2 + y 2 = 1; eliminating y, 
we have 
a 2 | 2 + b 2 (1 - f 2 ) - 2 (a 2 - 6 2 ) £a 3 + w 2 (a 2 - 26 2 ) + v 2 (b 2 - 2a 2 ) + 2 (a 2 - 6 2 ) v 3 V(1 - £ 2 ) = 0, 
or putting for shortness 
a 2 — b 2 = A, u 2 (a 2 - 2Z> 2 ) + v 2 (b 2 - 2a 2 ) = B, 
the equation for £ is 
A| 2 - 2A£m 3 + b 2 + B + 2Av 3 V(1 - F) = 0, 
but 
b 2 + B=b 2 (a 2 + r 2 ) + w 2 (a 2 - 2b 2 ) + v 2 (6 2 - 2a 2 ) = a 2 (a 2 - b 2 ) + v 2 (26 2 - 2a 2 ) = A (u 2 - 2v 2 ), 
viz. £ 2 - 2|m 3 + m 2 - 2v 2 + 2v 3 f(l - f) = 0, 
that is, (| 2 - 2m 3 £ + a 2 - 2v 2 ) 2 - 4r 6 (1 - £ 2 ) = 0, 
which is without difficulty reduced to the form 
(f — w) 3 — (a 3 — 3wv 2 )} = 0, 
that is, tj = u 3 — 3 uv 2 , 
and hence y = V s — 3va 2 , 
viz. writing u, v = cos 0, sin 0, then we have 
f = cos 3 9 — 3 cos 9 sin 2 9 = cos 30, 
y = sin 3 9 — 3 sin 9 cos 2 9 = — sin 30, 
or the circle of curvature at (acos0, &sin0) cuts the ellipse in (a cos 30, -b sin 30), as 
is known. 
C. XIII. 
68