139 
an optical deception, &c. 
y z= (^b — x). tang. X. 
where the co-ordinates are referred to the axis at right angles 
to the vertical generating radii, and passing through the cen- 
tre of their revolution : the basis b being measured on the 
axis from the point of its intersection with the curve to the 
centre : and x being the arc of the inner generating circle, as 
well as the abscissa.* 
A wheel simply rolling on its circumference exhibits, when 
seen through fixed bars, only those portions of the curves 
which are contained within the inner circle ; but when its 
motion of revolution is more rapid than its horizontal pro- 
gression, as when it is made to roll on an axle of less diameter 
on a raised rail- way, then the remaining portions of the 
curves will be seen, and others, on the lower part of the 
wheel, having a contrary flexure, will also make their ap- 
pearance. These are seen at FF in fig. 3. 
If the spokes, instead of being straight, be already curved, 
like those of the Persian water-wheel, their form, when 
viewed through bars, will undergo modifications, which may 
readily be traced by applying to them the same theory: 
Thus, by giving a certain curvature to the spokes, as in fig. 5, 
they will at one part of their revolution appear straight, 
namely, where the optical deception operates in a direction 
contrary to the curvature. 
The velocity of the apparent motion of the visible por- 
• This equality between the arc and the abscissa is a necessary consequence of 
the progressive motion of the wheel being equal to the rotatory motion of its cir- 
cumference : the former motion producing the increments of the abscissa; and the 
latter those of the arc of the circle. The equation (b — x). tang. x. is deduced 
from a simple analogy of the sides of similar triangles. 
