398 Mr Bennett, A Double-Four Mechanism. ~ | 
It may be found without difficulty that a vector from D equal 
to the vector function 
(DA’. DB’. D0’ — DA. DB. DC)(A A’ + BB’ + CC’) 
gives a point (44) satisfying all six conditions. Exchange of A | 
and A’ in this formula gives (11’), and similarly for (22’) and (33’), | 
The twenty-eight instantaneous centres are thus all eeepo ae. for, | 
and their collinearity in sets of three. 
§ 5. The double-four mechanism described so far is not the - . 
only one derivable from the symmetrogram of § 2. The point D | 
has, specialiy, been used as a centre for vector-products, and the | 
mechanism thus associated with D may be named (D). The use 
of D' in place of D gives a mechanism (D’) which is merely the | 
image of (D)in w. But three fresh pairs of mechanisms (A) and | 
(A’), (B) and (B’), (C) and (C’) complete a set of eight, consisting | 
of four distinct mechanisms (A), (B), (C), (D), each accompanied | 
by its image. These four may now be compared. 
Among the eight triangular plates of which (C) is composed | 
there occurs one whose vertices are the extremities of product- | 
vectors CA.CB, CB.CD, CD.CA. The sides of this triangle, as | 
vectors, are given by CD.BA, CA.DB, CB.AD; and this triangle | 
is identical im dimensions with the plate 4; and similarly for all | 
the rest. The mechanism (C) is composed of the same material | 
as (D), but it is differently put together. One of the isogramis © 
of (C) is obtained by multiplying the isogram ABA’B’ from C by | 
CD; and one of the isograms of (D) is got by multiplying the 
same isogram ABA’B’ from D by DC. These i isograms are there- 
fore congruent, with parallel sides and a half-turn rotation would 
bring them into coincidence. The triangles on corresponding | 
sides, moreover, are both congruent and homothetic. The other 
pair of congr vent isograms of (D) and (C) are got by multiplying | 
the isogram ABA’'B’ by DC’ from D and CD’ from C. Compare — 
with each the isogram af (C’) got by the multiplier C’D. The (C) 
and (C’) isograms, with the triangles on their sides, are images 
in «; and the (D) and (C’) isograms have the triangles on their 
sides congruent and howe anes and would themselves come to 
coincidence by a half-turn. There results the following method 
of converting the mechanism (D) into (C), namely :—() Separate | 
(D) into the two isogram mechanisms 13/24’ and 2’41’3. (a1) Ex- 
change two vertices of each plate 1, 3’, 2, 4° by giving it a half- 7 
turn Euout the middle point of the alle x the isogram on which 
it stands. (i) Exchange two vertices of each plate 2’, 4; 1 ae 
in the same way. (iv) Turn either of these new four-piece 
mechanisms upside-down ; i.e. give it a half-turn about some line 
in its own plane. (v) Unite the free vertices of 1, 3’, 2, 4’ with 
those of 1’, 3, 2’, 4 respectively. The double-four (C) thus formed 
