46 Proceedings of Royal Society of Edinburgh. [sess. 
This method also gives the co-ordinates of the vertex^ when we have 
J ax = ab{h a aos tii) ! c^ 
sjby = ab{a -f b cos o))/c^ . 
We may also find the vertex as the intersection of the axis and 
curve, when we get 
bx - ay =^\ 
2ab{bx -f ay) = -f cdW. ^ 
Thus 
^ab^^x — {ab -i- Xf = + a cos 
X = aP^if -i- a cos co)^c^ ; 
\a%y = {ab - Xf = ^a%\a + b cos 
y = a%{a -f- b cos (x))^/c'^. 
and 
