: 
Mr Bennett, A Double-Four Mechanism. 399 
has for its tetrads of plates 1, 2, 3, 4 paired with (and so not 
linked to) 2’, 1’, 4’, 3’ respectively. The double-fours (4) and (5), 
which may be similarly obtamed from (D), have 1, 2, 3, 4 
associated, the one with 4’, 3’, 2’, 1’ and the other with 3’, 4’, 1’, 2’. 
A double exchange of numerals suffices to convert the list of 
connections used for (D) into those necessary for (A), (B) or (C). 
Thus, in passing from (D) to (A), any vertex of 1, 2, 3, 4 is 
replaced by some fresh vertex in making attachment to any the 
same vertex of 1’, 2’, 3’, 4’; and the rule is to exchange the 
numerals 1 with 4 and 2 with 3. Since, ex gr., in (D) 12’ is 
attached to 2'1 (according to the original notation itself), so for 
(A) 43’ is attached to 21. The rule holds for all twelve pins. 
It may be observed that, taking the aggregate of all four 
mechanisms, any particular plate 1, 2, 3 or 4 is connected in 
turn with all the twelve vertices of the plates 1’, 2’, 3’, 4’, and 
conversely. The two tetrads remain distinct throughout, and 
connections occur only between members of opposite tetrads; but 
the different pairings which distinguish the double-fours are 
peculiar to each mechanism in turn. The mechanism given in 
Fig. 3 is the mechanism (D) derived from the symmetrogram of 
Fig, 2. 
§ 6. Some of the special forms of the isogram double-four 
mechanism (D) may be briefly noticed, arising from special forms 
of the symmetrogram from which it is derived. 
G) IfA,8B,C, Dare concyclic, the triangular plate 4 becomes 
a straight bar. 
(i) If A, B,C, D are concyclic, and also A’, B’, C, D, then 
both plates 3 and 4 become bars. 
fan) liye sl BT CD and (Al B.ClD) and A’, BC, D are 
concyclic, then plates 1, 2,3 are bars. In this case the 
symmetrogram is obtainable from the figure of any 
triangle A BC, with A’, B’, C’ as the feet of its perpen- 
diculars, and PD as orthocentre, by inverting from any 
. point on the polar circle. 
(Gv) If A, B,C, D are on a circle with diameter , then 
A’, B’, CO’, D’ are also on the circle. All eight pieces 
are then bars, and the axes of the three pairs of 
isograms are concurrent. The figure occurs in a 
paper of the author's (Lond. Math. Soc. 1911, Ser. 2, 
Vol. x. p. 383). The eight points of the symmetro- 
gram of Fig. 2 are approximately concyclic; and as 
a consequence the plates of Fig. 3 take an elongated 
form differing not greatly from bars. 
(v) If D and D’ coincide, the double-four is symmetric 
about #, and pairs of its plates are images in 2. 
(vi) If D and D’ are coincident in case (iv) the eight-bar 
