113 Dr, lloget*s Explanation of dn Optical Deception. [AvUp 



Bach aperture produces its own system of spectra ; and hence, 

 when the apertures occur at short intervals, the number of the 

 «pokes is considerably multiplied ; but if the intervals be so 

 adjusted as to correspond with the distances between the spokes 

 at the circumference of the wheel, the images produced by 

 each aperture will coalesce, and the effect will be much height- 

 ened. 



A mathematical investigation of the curves resulting from the 

 motion of the points of intersection of a line moving parallel to 

 itself, with another line revolving round its axis, will show them 

 to belong to the class of Quadratrices, of which the one which 

 touches the circumference of the inner generating circle is that 

 which is known by the name of the Quadratrix of Dinostrates^ 

 Such a system of curves is represented in fig. 3, where MC, 

 CN, are the generating radii, A the outer, and B the inner gene* 

 rating circles, and PQ the common axis of the curves. 



All these curves have the same general equation, namely, 



y -= {h "- X . tang, x, 



where the co-ordinates are referred to the axis at right angles 

 to the vertical generating radii, and passing through the centre 

 of their revolution : the basis h 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 progression, 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 appearance. 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 operat(;s in a direction contrary to the curvature. 



The velocity of the apparent motion of the visible portions of 

 the spokes is proportionate to the velocity of the wheel itself; 

 but it varies in different parts of the curve ; and might therefore, 

 if accurately estimated, furnish new modes of measuring the 

 duration of the impressions ofhght on the retina. 



* This equality between the arc and the abscissa Li a necessary consequence of the 

 progressive motion of the wheel being equal to the rotatory motion of its circumference : 

 the former motion producing the increments of the abscissa ; and the latter those of the 

 •re of the cjrcle. The equation y = (6 — jr), tang. x. is deduced from a simple ana* 

 logy of the tides of tomilar triangles. 



