| Mr Bennett, A Double-Four Mechanism. 395 
Let the lines joining D to the six points A, B, C, A’, B’, C’ be 
taken as vectors, and let all the vector-products of these be taken 
in pairs, omitting only the products DA.DA’, DB.DB’, DC.DC". 
The extremities of these twelve vectors give a figure of twelve 
points. The vector DA.DB’ gives a point 12’, DA’. DB gives a 
: point 21’, and so on, in accordance with the tabular scheme 
1’ Oy 3/ 4! 
1 af. AB' O'A BCL an | 
2 A'B es BC’ CoM 
3 CA’ BC cs A'B' 
4 BC CA AB es 
where, on joining D to the extremities of any line of the symmetro- 
gram entered in the table, the vector-product is to have its 
extremity named by the numerals of the same row and column. 
(It may be understood that an arbitrary unit vector divides all 
the products in common, maintaining the vector dimensions 
correctly and giving an arbitrary scale and orientation to the 
resultant figure.) 
Consider the triangle formed by the points 41’, 42’, 43”. Its 
sides 41, 42, 43, as vectors, are given by the differences of the 
vector-products DB.DC, DC.DA, DA.DB, and are therefore 
equal to DA.BC, DB.CA, DC.AB (with zero sum). The lengths 
of these vectors have appeared as constants in § 2, and hence the 
triangle 41’, 42’, 43’ has sides of constant length. Similar results 
hold for each of eight triangles 1, 2, 3, 4, 1’, 2’, 3’, 4’. The sides 
of each triangle have lengths equal to those of the vector-products 
of pairs of sides of an associated quadrangle of the symmetrogram ; 
the four points consisting of the three points named in the row 
or column of the above table, together with D. A mechanism 
is therefore obtained, of the double-four type, with a notation 
corresponding to that of Fig. 1. Further, the three pairs of 
deformable quadrilaterals are all isograms. One is derived from 
the isogram ABA’B’ by multiplying it, from D as centre, by the 
vector DC; and the companion isogram is got by multiplying 
by DC’. These are both similar to the isogram ABA’B’; the 
ratio of their linear dimensions and the angle of inclination of 
their axes being given by the ratio and inclination of the vectors 
DC and DC’. A list of the lengths of the twelve pairs of equal 
lines of the mechanism figure is as follows: 
