Mr. Burch's Method of Drawing Hyperbolas. 377 



The Use of the Quadruplane as a Hyperbolagraph. — The 

 two asymptotic coordinates of any hyperbola evidently form 

 two sides of a parallelogram of constant area. Hence any 

 hyperbola can be readily traced by the use of the Sylvester- 

 Kempe quadruplane linkage, the four vertices of which lie at 

 the four angular points of a parallelogram of constant area 

 and constant obliquity*. 



The product of the adjacent sides of this parallelogram, or 

 as Sylvester calls it the " modulus " of the cell, is equal to 



B2 _ A2 sin aiS in«. 2 = 

 sin^tf 



where B and A are the distances between the pivotal points 

 of the long and short links of the cell, a x and a 2 the angles 

 adjacent to the line joining the pivotal points, and the angle 

 subtended by this line at the intervening vertex. If we make 

 each link of the cell symmetrical we have 



ai = « 2 = 90-19/2, 



and M _ B 2 -A 2 



" 4sin 2 0/2- 



The obliquity of the parallelogram is equal to 0. 



Hence to describe an hyperbola whose axes are a and b we 

 must make the modulus of the cell equal to the modulus of 

 the hyperbola, or 



4 

 and the angle equal to the angle between the asymptotes, or 



2ab 



= sin' 



a 2 + 6 2 



Then if one vertex of the cell is fixed and an adjoining 

 vertex is moved along a straight line (the edge of a T-square 

 or straight edge for example), passing through the fixed 

 vertex, the vertex diagonally opposite the latter will describe 

 the required hyperbola, having the fixed point as origin and 

 the straight line as one of the asymptotes. To describe 

 different hyperbolas it is necessary to be able to vary both 

 and M. The first may be easily done by making each link 

 in two halves, pivoted together at the vertex of the link with 

 a divided sector and clamp, by means of which the desired 



* See " The History of the Plagiograph," Sylvester, ' Nature/ vol. xii. 

 p. 214 ; also Kempe, ' Lecture on Linkages/ p. 25 et seq. 



