Daedaleum, a new Instrument of Optical Illusion. 39 



where, ■& remaining constant, <p will vary, differentially, as \J/ 

 varies. The greatest effect of the neglected portion occurs 

 when d = a\/3, which is therefore the worst situation for ob- 

 serving the phenomenon. 



(3.) If the eye is brought close to the revolving cylinder, 

 d becomes = a .:. <p — 3, or each projected division occupies 

 exactly the same breadth as each of the originals. 



• r- • a 



(4.) If the spectator removes to an indefinite distance, —r 



approaches as its limit .'. $ = S — \J/ or 7r + 3— \f/; and <p 

 varies differentially as 3. Therefore the projected divisions 

 will be exactly equal to each other. As these last conditions 

 are those which a lecturer will require, artists should keep 

 them especially in view. And since in this projection all the 

 divisions round the outer circle are projected upon just a semi- 

 circle, all figures and objects depicted should be considerably 

 exaggerated in breadth, nearly in the proportion of 2 to 1 when 

 compared with the natural breadth. 



(5.) Objects connected with the effect of those in the cylin- 

 drical border may be drawn in corresponding sectors on the 

 disk itself. But the projections of these sectors will of course 

 appear considerably curved if extended quite to the centre. 

 The curves produced, however similar in appearance to those 

 observed by Dr. Roget, are, notwithstanding, of quite a di- 

 stinct class. A few of their properties it may be worth while 

 to unfold. 



(a.) If \J/ = + — , u = f — — - gives the two points 



2» cos +j 



where the tangents ET, E* are likewise tangents to the curve. 



, ., . _1 a sin 3 



(/3.) If \J/ = 3 or 7T + 3, u = oo while <p = + sin — - — ? 



indicating two asymptotes, which are tangents from E to the 

 circle whose radius is sin 3. 



(y.) If <p = or 7r, .\ u = —d, x = — d, and y = 0, indi- 

 cating a nodus at E. 



The entire curve has then four branches of the hyperbolic 

 species, two of which have double curvature, touching given 

 lines, and then returning to become asymptotes to lines in- 

 terior to those. 



When E is at an infinite distance the curves are of a more 

 simple kind, and the family to which they belong becomes ap- 

 parent. The most simple of all is when S = ± ir, giving 



a sin Jr . , , , • tu • 



x — ■ . , or a sin J/ tan vp and y — a sin v{/. hliminating 

 cos \p 



