70 
Proceedings of the Koyal Society of Edinburgh. [Sess. 
or 
where 
D = 
D + Er sin <f > , 
and E = o>— 
[X /x- 
Then the condition for a brachistochrone is first evidently that the- 
path shall lie entirely in the plane of rotation, and further 
8 f — =0, 
J D + E?’ sin (j> 
or, neglecting the second order of small quantities 
\ f ds j, f i 
dsEjr sin <f> 
= 0 . 
If we consider the part of the brachistochrone intercepted between 
A and B, then Jcfe is the total length of the curve AB, while Jds r sin 0 
is twice the area traced out by the radius vector. 
It is easy to see that, while an increase of the length of the path leads 
to an increase in the time taken, an increase in the area covered by the 
radius vector will have the opposite effect, if area be reckoned positive 
when swept out in the direction in which the solid is rotating. The last 
equation may be interpreted to mean that for any slight deviation from 
a brachistochrone these two causes of variation must exactly balance, and 
that they will do so if an increase in length is accompanied by D/2E times 
the same increase in area. Substituting we find 
D fxc 
2E == 2o i (^-l)' 
It is easy to show that a curved path, with a certain radius of curvature, 
constant within the limits already assigned, will satisfy the above condition. 
In the first place, it is a well-known theorem, and may be proved by the 
