392 Mr Bennett, A Double-Four Mechanism. 
when the plates are assembled as a mechanism. The side of 
plate 1 opposite to vertex 12’ will be denoted by 12. These rules” 
of notation are to be followed for all the twenty-four vertices and 
all the twenty-four sides. (The numerals assigned to any vertex — 
have always one accented, and those assigned to any side have both — 
accented or neither.) 
If the pins at points 12’, 21’, 34’, 43’ are removed, the whole 
divides into separate mechanisms. One of these consists of plates 
1, 3’, 2, 4’ consecutively linked at the vertices of a deformable 
quadrilateral which may be denoted 13’24’. Its sides are 12, 3’4’, 
21, 4’3’, and the free vertices of the four triangles form a quadrangle 
12’, 3’4, 21’, 43. The other mechanism consists of plates 2’, 4, 1’, 3 _ 
consecutively linked at the vertices of a deformable quadrilateral 
2’41’3 with sides 2’1’, 43, 12’, 34; the free vertices of the triangles 
forming a quadrangle 2’1, 43’, 1/2, 34°. The two quadrangles are 
congruent, and coimcide when the mechanism is reassembled. 
Such a division into two parts may be effected in three different 
ways; and there are thus three such pairs of quadrilateral linkages. 
Mechanisms of the type described are singular in possessing one 
degree of freedom; for the connectivity gives, normally, fourfold 
stiffness. They were first discussed by Kempe (‘‘ Conjugate four- 
piece linkages,’ Proc. Lond. Math. Soc. 1878, Vol. 1x. pp. 188— 
147), who gave five different species, and afterwards by Darboux 
(“Recherches sur un systéme articulé,” Bulletin des Sciences Math. 
2 série, t. 11. 1879, pp. 151—192), who carried out an exhaustive — 
analysis and completed the catalogue by the addition of a species 
in which the three pairs of quadrilateral linkages are all contra- 
parallelograms (here called isograms). He shows that the material 
of the mechanism depends upon six parameters only, and that the 
three pairs of isograms are similar in pairs; but, for the rest, 
leaves the figure dependent for its precise description upon 
thirteen simultaneous equations in complex variables. It is this 
mechanism which is to be further discussed here. As a con- 
sequence of some investigations the following cardinal properties 
offer themselves. (a) The centre of similitude for each pair of 
isograms is the same. (8) The feet of the perpendiculars from 
this centre upon the axes of symmetry of any two pairs of isograms 
are the vertices of an isogram. (vy) This isogram is similar to 
the third pair of isograms of the mechanism, and with the same 
centre of similitude. These results being found, a simple means 
arises of constructing the mechanism ab initio. As it has been 
somewhat elusive, and as its geometry is a little uncommon, kine- 
matically, there seems sufficient reason for a short study such as 
is here presented. 
2. An auxiliary diagram (Fig. 2), not itself a mechanism, 
may be described first. It consists of an axis of symmetry 2, 
ot i 
