An elementary treatise on the differential calculus, in which the method of limits is exclusively made use of, by the Rev. M. O'Brien.

XI SCELLA,NEROT S TIE OREMS. 193
also, sin 0 = - a - y.. i =vers- = s /2ay - y 2........ (*),
a
which is the equation to the cycloid,
The point B at which 0 = wr, which is called the vertex Equation
of the cycloid, is often taken for the origin, the line BY' tavered
parallel to AX for the axis of y, and the line BX' perpendicular to AX for the axis of x. We may easily find
the equation to the cycloid referred'to these axes by putting
in the equations (1) y'+ AX' or y'+ 7ra for x, and BX'-a/
or 2a- x' for y, which gives us
y + 7ra = a (0 - sin 0),
2 a - v' = a ( - cos 0);
which, if we put 7r + 8' in the place of 0, become
y'= a (0' + sin 0) '
(l (1- cos ')
and therefore as before (suppressing the dashes),
y = a vers -1- + 2ax - x2......... (4),
a
which is the equation to the cycloid referred to the vertex.
To determine the tangent and normal at P. Tangent or
normal to
a cycloid.
From the equations (1) we find
d = a (1 - cos 0) dO, dy = a sin Od;
dx a (1 - cos 0)
dy a sin 0
PM
- = tan PNX.;
MN
1.3