74 
Proceedings of the Royal Society of Edinburgh. [Sess. 
of intersection of that axis; C the projection of B on P. Join CA and AB, 
and let \fs 0 be the angle CAB. Let ADB be a possible path for light 
between A and B, and let AEC be its orthogonal projection on P. Let 
\]s be the angle between any element of the curve and the corresponding 
element of the projection; r the distance of the latter element from the 
point O, and (p the angle it makes with the direction of r produced. Then 
the speed of light at any point on ADB is 
c • , , fX 2 - 1 
- + cor sin <jt> cos if/ ™ - , 
/x ^ 
or 
D + Er sin <£ cos if/ , 
where D and E have the same meanings as before. The condition for a 
brachistochrone is therefore 
v fds s f 
‘ h- s ) 
Er sin cos i J/ds 
IP 
If now ds' be the element of the projection corresponding to any element 
ds, ds ' = ds cos \fs, and our condition may be written 
s fds « fEr sin d>ds A 
s Jd- s ]—w - =0 
Let Q be the cylindrical surface whose generators are parallel to LM, 
and which passes through the curve ADB and its projection. Then it is 
evident that for any curve on Q joining A and B the latter half of the 
expression vanishes. Hence the curve of this class which most nearly 
satisfies the brachistochrone condition is the shortest in length, or otherwise 
the curve for which \{s is constant. 
We may therefore confine ourselves to curves for which this condition 
is fulfilled. Denoting the length of the arc AEC by l, and that of CB by a, 
tan ^ = T’ and C0S ^ = -Jiha?’ 
The condition may therefore be written 
As 
fds' _ 
JT~ 
1, we get 
g fds Jl 2 + a 2 _ « f Er sin <pds' _ ~ 
J D^ J W 
Ihl ^ f E?’ sin <pds _ n 
or in other words, being the rectilineal angle CAB, the increase of area 
subtended at O by the curve AEC must be ^ C ^- r ^ Q times its increase of 
length, to the first order of small quantities. From this it follows that 
