1883.] On Carves circumscribing Rotating Polygons. 



323 



If, however, without confining ourselves to necessarily small values 

 of e, we consider only the first term e 1? which is always the most 

 important in practice, the curve • • • is a hypotrochoid having 



for its Cartesian coordinates 



y=p sin — e sin w0, 



x=p cos + e cos ncfi, 



where, if a and b are the radii of the rolling circles, 



b=?±. 



n 



Thus we see that a hypotrochoid with its generating circles in the 

 ratios given will circumscribe an w-sided rotating polygon, the dia- 

 meter of whose circumscribing circle is 2p Q . In order that the rota- 

 tion may be mechanically possible, the polygon and hypotrochoid 

 must not cut one another at any part of the revolution, and this 

 condition limits the possible value of e. 



Let ABCD, fig. 4, be part of the polygon, and EFGrH part of the 

 hypotrochoid, then it may be shown that if a side of one figure cuts 

 that of the other, the cutting sides will contain the greatest area 

 between them when, as at B and 0, the adjacent angles of the polygon 

 are equidistant from the adjacent corners of the hypotrochoid. For 

 the angular motion of B and C is a maximum when passing the 

 middle of the side of the hypotrochoid, and a minimum when passing 

 the corners, so that, since BF = CGr, C will leave Gr faster than B 

 approaches F, and vice versa, hence the area enclosed between the 

 points p 2l by BC and FGr will diminish if the polygon rotates, and 

 hence if BC is a tangent to FGr when BF = CGr, the sides can never 

 cut one another. The distance of the middle point of the side of the 



polygon from its centre is p Q cos -, while the distance of the corre- 



n 



sponding point of the hypotrochoid is p — e, and equating these quan- 

 tities, we get as the greatest admissible value of e, 



e= Po ver s -. 



n 



If the hypotrochoid is to be everywhere concave to the centre, e must 

 not be greater than 



The following table shows the values of e in the two cases from 

 n=2 to n= 7 : — ■ 



