Mr Bennett, A Double-Four Mechanism. 397 
form, the relative rotation of adjacent sides being four right 
angles. Among the forms occur two in which the vertices and 
‘sides are all in one straight line; and the original form itself 
appears in all four times, on two pairs of occasions which alternate 
with the rectilinear forms. This simultaneous cyclic performance 
of all the six isograms of the mechanism may be followed by 
observing the effect on the symmetrogram of inverting from the 
travelling pot 0. As regards the isogram ABA’B’, it inverts 
into collinear points when O crosses the circumference of the 
‘circumcircle; and it inverts into an isogram similar to ABA'B’ 
when 0 is at either diagonal point V or N’, or at the centre 
of the circle, or at infinity. Moreover the two isograms 1'32’4 
and 24/13’, similar to ABA’B’, will have parallel axes when O 
erosses the circumference of the circle circumscribing CDC’D’; 
and when O passes the centre of this circle the axes are inclined 
at the same angle as originally (with O at infinity). 
As regards the kinematics of the instantaneous movement, 
any two of the eight pieces have a centre of relative rotation, 
and the three centres associated with any three pieces, taken in 
pairs, must be collinear. The figure necessarily possesses the 
requisite collinearities, and they may be readily accounted for. 
Let the instantaneous centre for plates 1 and 2 be denoted 
by (12), and similarly for all others. Of centres such as (12’) 
there are twelve, these being permanent centres given by the 
connecting pins themselves. Of centres such as (12) and (1’2’) 
there are altogether twelve. The centre (12) is collinear with 
(13’) and (23’), and is also collinear with (14°) and (24’); and 
similarly for all such others. For any pair of equal sides of an 
isogram, that is, the plates they carry have as instantaneous 
centre the (diagonal) point of intersection of the other two equal 
sides. 
| Of sets of three plates there occur three different types, of 
which 123’, 1’2'3’ and 144’ may be taken as representative. For 
the first, the collinearity of (12), (18’) and (23’) has already been 
noticed. For the second, the collinearity of (1'2’), (2’3’), (81) 
“may be seen thus. The vector from D to (1’2’), a diagonal point 
of the isogram 1'32’4, is given by the product-vector DC.DJN, 
where WV is the intersection of AB and w; (2’3’) is given similarly 
by DA. DL and (3'1’) by DB. DM. These are the three products 
of pairs of vectors drawn from a point D to the three pairs of 
vertices of a complete quadrilateral, formed by the sides of the 
triangle ABC and the line 2; and hence the extremities of the 
three product-vectors are collinear. 
There remain only the points (11’), (22’), (83’), (44) to be 
considered. The last should be collinear with three pairs such 
as (14) and (14’), and also with three pairs such as (14) and (1’4’). 
