32 
Proceedings of Royal Society of Edinburgh . [ dec . 20 ’ 
curve, where OC = 2a, touching the curve at the point r — 00 , 0 = -k. 
We are concerned with values of 0 only from 0 to tt. We can 
readily show that the curve BD touches the circle at B. For in 
(1) putting a + Sr for r, and J7 r + 80 for 0, we get 
s*-£sr-w( i L + ^ + ^ +& c.) = ° ! 
or Sr = 0. If we retained terms involving SO 2 , we could readily 
prove that a is the radius of curvature at B, and that C is the 
centre of curvature, so that there must he a point of contrary flexure 
between B and D. As before remarked, DBA# is a stream line. 
Thus, as we pass along DB, (1) is its equation. When we get on 
to the circle BA, we shoidd get, putting r — a, 
2 2 2 
0 - 1 7r sin 0 + s * n ^ ~ 3 4 ~ 5 s ^ n ^ + "5^6" 7 s * n ~ &c. = 0 > 
and this shoidd be true from 0 = 0 to 0 = |7r. But if we were to 
expand 0 - sin 0 by Fourier’s theorem in a series of sines of even 
multiples of 0 between the limits 0 and Jvr of 0, we should get 
exactly the above series, which verifies the work, and we may 
observe that the expansion would be true even at the limits, for 
0 - sin 0 vanishes when 0 = 0 and 0 = J7r. 
It will be found on examination that if AB, instead of being a 
quadrant, were any part of the semi-circumference, a similar pro- 
position would hold good. 
It can also be shown that if AB, instead of being a quadrant of a 
circle, were a quadrant of an ellipse, a like proposition could be 
established. I will give a slight sketch of the proof. Suppose AB 
to be part of an ellipse, and suppose it to be moving parallel to OA 
with a velocity V. Let u, v be the resulting absolute velocities of 
the fluid at any point (x, y). Then we may take 
dd> dch d 2 (b d 2 d> 
W = dx’ ‘' = dy’ + • < 2 ) 
Now put x = c cosh /3 cos a, y — c sinh /3 sin a, 
where c is a constant, sinh and cosh are the hyperbolic sine and 
cosine, and a, /3 are new so-called curvilinear co-ordinates. Then 
(2) becomes 
d 2 <f> d 2 (f> 
