400 Mr Bennett, A Double-Four Mechanism. 
mechanism becomes symmetrical. This is the figure | 
given, apparently, by Darboux (loc. cit. p. 174). / | 
(vii) If D and D’ are at infinity on «, the product-vector | 
extremities may (as a limit) be taken at the middle 
points of the sides of the isograms ABA’B’, BCB’O’, | 
CAO’ A’. The mechanism is the same as in case (Vv), | 
but at a different stage of its deformation. Since | 
the plate given by the middle points of triangle ABC | 
is similar to ABC and of half the size, and similarly | 
for others, it follows that the material for such a — 
mechanism may be supplied by the eight triangles | 
whose vertices are A or A’, B or B’, C or C’ in the } 
symmetrograin. 4 
(viii) If Cand C’ coincide with D and D’, the isogram 1'324 
of (D) becomes evanescent in size, and the pieces © 
1’, 2’, 3, 4 become bars linked together at a common 
extremity. 
In all these cases peculiarities affecting the mechanisms (A), | 
(B), (C) will accompany those of (D). : : 
| 
Nore. Fig. 1 has served a purely schematic purpose in the 
foregoing treatment of the isogram double-four: its three pairs — 
of quadrilaterals are drawn as parallelograms instead of isograms. 
But, as a consequence, it is itself a double-four mechanism of 
another species, and is included in his list by Darboux (loc. cit. 
p. 164). He identifies it with a last case of Kempe; but with 
some confusion, for the latter has a connectivity different from 
that of the double-four type. The freedom of the mechanism is a 
simple consequence of the parallelogram construction. It may be 
noticed that if the pins are removed, the set of plates 1, 2, 3, 4 
may, without rotation, be brought together so that pairs of equal - 
sides coincide, the vertices of the four triangles forming a quad-_ 
rangle; and similarly for 1’ 2’ 3’ 4’. So that the material for the — 
mechanism may be supplied by the eight triangles givensby two — 
arbitrary quadrangles. (In Fig. 1 these quadrangles are made, 
unessentially, and for simplicity, two equal parallelograms.) 4 
But this mechanism, though ostensibly a double-four, should — 
in strictness be classified as spurious or improper; for the pairs of — 
pieces that are not linked have centres of relative rotation which — 
are not variable but permanent, and for which the missing pins of 
connection may be supplied. The centre for 1 and 1’ is such that 
it completes the figure of the first quadrangle when associated — 
with the vertices of 1; and with the vertices of 1’ it forms the ~ 
figure of the second quadrangle ; and similarly for the other three — 
pairs. The completed mechanism consists then of four rigid 
quadrangles linked by their points to four other quadrangles; 
Poy yan) Cal ee 
