36 Mr. W. G. Horner on the Properties of the 



This angle, which can only be known by experiment, let us 

 call «. As AE, EF is the shortest course from A to F, if the 

 angle AKO be «, the angle ELW is ?r — «. The same may 

 be said of EF, FG, the shortest course from E to G, and of 

 FG, GB, the shortest course from F to.B. If AE, BG be 

 produced to meet in C, FC is plainly a parallelogram, and 

 the brachystochronous course from A to B is also AC, CB. 



From all that precedes we may derive the following simple 

 construction for the brachystochronous course, when the 

 angle a. is known. Let it be required to sail from A to B, 

 (fig. 2 ) in the shortest time possible, the wind blowing from 

 B towards N. Upon AB describe a segment of a circle AZB, 

 containing an angle equal to ir — 2a. Complete the circle 

 and bisect the arc AVB in V. Draw VZ perpendicular to 

 BN and join AZ, ZB. Then AZB is the course required; 

 for as AZB = v— 2a, each of the angles ZRB, ZBR is equal 

 to a. 



Papworth St. Everard, Dec. 13,1833. 



XL On the Properties of the Daedaleum, a new Instrument of 

 Optical Illusion. By W. G. Horner, Esq.* 



r |' , HE ingenious and amusing invention of M. Plateau, by 

 ■*- superadding the resources of art to those of science, has 

 rendered an instructive experiment exceedingly popular. But 

 neither the Professor nor any of his imitators have added any- 

 thing to the mathematical principles, which remain hitherto 

 in exactly the position in which Mr. Faraday left them nearly 

 three years ago, in the Journal of the Royal Institution. The 

 repose of one portion of the spectrum, the residual motion 

 apparent in the advance or retrogression of others, and the 

 blending of variation of action with identity of subject, have 

 been traced to their causes, both by Mr. Faraday and Dr. 

 Roget, most satisfactorily ; nor does it appear that any phaeno- 

 menon observable in the relative motions of a wheel and a 

 system of detached bars, or of a pair of perforated disks, has 

 escaped the notice of one or other of those gentlemen. One 

 set of phasnomena, derived from a still more simple ap- 

 paratus, has, however, been left unnoticed, as far as I can dis- 

 cover, by all ; and my design in submitting the present paper 

 to the readers of the Philosophical Magazine and Journal of 

 Science is to familiarize both the principle and the exhibition 

 of an experiment involving all the interesting illusions of the 

 phantasmascope, but capable of being performed without a 



* Communicated by the Author. 



