223 
of Edinburgh, Session 1867-68. 
culars from S on the tangents at P and p, p and p' the radii of 
curvature at P and p, then 
r r 1 
~ ~ i • 
ZXT 
Also the velocity at p is 
II = 
v 
r' t 
But, since we have II = r' . — . - 
p w 
(as we see by expressing it in terms of the angular velocity of Sp), 
if Sp' be called r\ we have 
Hence, as 
nir = -v 
tt/ ,f 
II = r 
r 
* P 
r 
3T 
zxy' = A , 
t n 7/ 
■srr = A , 
A rV' 2 
A 
r'A' 2 
A' 3 
r 2 * & ~ 
r 2 
'3 ~ 
A 2 
r r 
Or, more simply, if v be the velocity in the orbit, we have, by 
expressing the centrifugal force in terms of the normal component 
of the acceleration, 
'O' ZtT 
-= n- . 
p r 
Hence 
[This is the well-known formula 
]i 2 dzr _ 
n •] 
Thus 
because from 
AV A'V A' 2 , 
nn — ' n 
TXT p ‘Zt p 
TtST — r'zv = A 
we have at once rV 2 = zr&'pp' . 
4. Again, if the hodograph be a circle described with uniform 
angular velocity about a point in its circumference, the path is the 
cycloidal brachistochrone. 
