44- SCIENTIFIC APPARATUS. 



we may use the old-fashioned elliptic trammels, or some other 

 form of elliptic compasses of more recent invention. If none of 

 these arrangements are at hand, perhaps . the best way to obtain 

 easily a really good ellipse is to take an oblique section of a 

 carefully turned cylinder. All the three conic sections can be 

 described by means of the conograph of Dr. Zmurko, of Lem- 

 berg. 



(2.) The cycloid is the curve traced by a point in the circum- 

 ference of a circle rolling on a straight line ; the epicycloid and 

 hypocycloid are traced in the same way, only that the rolling circle 

 rolls on the outside or inside of a circle instead of on a straight 

 line. These definitions have suggested various modes of describing 

 these curves mechanically ; an interesting cycloidograph is exhi- 

 bited by Dr. Zmurko. 



(3.) Mr. A. G. Donkin has constructed a beautiful apparatus for 

 tracing harmonic curves. This machine will draw as many dif- 

 ferent forms of the curve y = a sin (m x + a) -f- b sin (n x + fi) 

 as there are means for varying the constants a, b, m, n, a, /? : the 

 number of variations being practically unlimited, except in the 

 case of m and n, which are the numbers of teeth in certain 

 wheels of which only a limited number of changes can be pro- 

 vided. Thus the machine exhibits to the eye the effect of the 

 composition of two harmonic curves of any different intensities 

 and phases, and of different intervals. 



(4.) Several forms of spirals (or volutes) are of use in the arts, 

 and appropriate modes of describing them have besn given. 

 Among these curves we may select for special notice the involute 

 of the circle, which gives the proper form for the teeth of toothed 

 wheels. 



(5.) We may refer, lastly, to the epicycloidal curves of Mr. 

 Perigal ; and to the beautiful diagrams, not properly epicycloidal, 

 but of a more complicated type, obtained by his compound 

 geometric chuck. 



