560 • REPORT — 1894. 



together at one end, 0, and jointed at their other ends to two of the points A. 

 These bars OA may be called radius bars, the other bars, AD, being called lonff bars, 

 The proof is easily gathered from an inspection of the accompanying figure, in which 

 OA„ OA,, OAj, OA^ are radius bars, AjD, A^D long bars, and A,Cj, A3O, portions 

 of two other long bars whose remaining portions are indicated by dotted lines. The 

 figure contains two equal and similar jointed rhombuses, OCj, OC3, and three 

 similar kites, OB, DB, DO. Each of the bars Afi^, AgCj is cut in a fixed ratio at 

 the pohit of crossing, B, the ratio of the smaller part to the whole being OAj : A J) ; 

 bence we can have joints both at B and D, as well as at 0, and the other corners 

 of the rhombuses, without hampering the motion. 



Two radius bars are in general sufficient to give the centre, but we may in 

 theory attach a radius bar at each point A, and joint their other ends together at 

 one point, 0, which will be the common centre. There are difficulties in the way 

 of realising this design in practice, except with a very limited range of movement ; 

 but by carefully selecting the best order of superposition of the bars, and by 

 thinning oft' the radius bars towards the end where they are all superposed, it has 



been successfully carried out in two of the frames exhibited, each consisting of ten 

 long bars and ten radius bars. The other frame exhibited illustrates the fir.-t 

 paragraph of this abstract, and consists of ten bars BD. The number of bars 

 in this frame might be increased indefinitely. 



If wi denote the ratio of a long bar to a radius bar, or of the longer to the 

 shorter sides of a kite, 2 a the angle between the two shorter sides, and 2 /S the 

 ano-le between the two longer sides of a kite, then, by considering one of the two 

 triangles into which a kite is divided by its axis, we have sin a/'sin /3 = m, which 

 is equivalent to 



tan i (a -/3)/tan i(at^) = {m- l)/{m + 1). 



a + 3 is one of the angles of a rhombus, and a-^ is the angle between two con- 

 secutive rays of a star. , ., n 



In the case of the frame first described, consisting of 2w bars BD, when a 

 Tegular star of n rays is formed, the central figure in the frame will be' a polygon 

 of 2 T» sides, whose angles are alternately a + ^ and 360° -2 a. 



