396 Mr Bennett, A Double-Four Mechanism. | 
23=|DA'.BC’| 14=|DA.BC 
31=|DB’.CA’| 24=|DB.CA 
12=|DC’.AB’| 34=|DC.AB ! 
yy =\Da Bo"! * V4 = pa net iar (ra) 
3/l'=|DB.CA'| 4° =| DB’.OA 
12) 0 84 sc nea 
Six relations should connect these twelve lengths. It appears : 
immediately that : 
giving three relations. Further (using lengths and not vectors) 
1232 — 232 = (DA? — DA”) BO”? = AA’. DD’. BO”, 
14? — 14? = (DA? — DA”) BC? =— AA’. DD’. BC?, 
and hence 
(232 + 14) — (2'3" + 1’4) = AA’. BB’. CO’. DD’ 
the constant of § 2, giving two further relations. The sixth 
relation is supplied by the determinantal equation (v) on putting | 
for ky» its value in terms of the sides given by (iii), and by 
AB/AB' = 34/12’ = 34/12 jointly; and similarly for the other 
elements of the determinant. 
The angles of the two sets of plates are also simply related. 
The angle of plate 1 opposite to the side 12 may be denoted 
by 12, and similarly for the rest. Then at the point 12’ the sum 
(or difference) of the angles 12 and 2’1’ is equal to the sum 
(or difference) of the angles of two of the isograms; and this 
same sum (or difference) occurs for the angles at the points 
21’, 34’, 43’. Similar results hold for the rest of the angles. The 
angles of the plates 1, 2, 3, 4 thus serve to determine those of the 
other set of plates; and one method of derivation may be put 
thus :—Let angles a, 8, y be taken such that 
Qa = 14 + 23 + 32 + 41, 
28=13 + 24431 +49, 
My = 12 + 214344 43. 
Then on subtracting from a, 8, y the angles of any one plate 
(those namely which occur in a, 8, y respectively) the angles of a 
plate of the second set are obtained. | 
§ 4. <A deforming isogram, starting from any arbitrary form, 
may pass through a cycle of fresh forms and revert to its original 
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