394 Mr Bennett, A Double-Four Mechanism. 
is identically true for all positions of P. Taking P at A, 
a.AA?+b(AB?+ AB”) +c(AC? + AC?) 
+ d@(AD?+ AD?)=0 (11), 
and putting (A B/AB’Y = (ks — 1)/(ks + 1), 
so that kio= (4B? 4 AB?) AAS BB OR eer (iii), 
(11) becomes 
a.AA’+b. BB’ .ky+c¢.CC’.ks+d.DD’'.k,,=0...(iv). 
By taking P in turn at B,C and D three other such equations are | 
obtained, and hence by elimination 
po (be (By (Bu 
Ks, I kos Feng | 
: |= QO) eee 
ee ian ©) 
| ky ky Ky 1 | 
The six ratios, such as A.B/AB’, remain all constant, therefore, if 
any five are kept constant; and the figure may therefore assume | 
a single infinity of different shapes while each of the six isograms _ 
retains a constant value for the ratio of its sides. 
The variation of shape may be obtained from any arbitrary 
initial form by inverting the figure of eight points from a varying 
pomt O on the axis of symmetry z If the radius of inversion 
is f, the lengths AB and AB’ become, after inversion, 
R?. AB/OA.OB and R?.AB’'/OA.OB’, 
the ratio of which is AB/AB’, the same as before. Inversion 
leaves each of the six isograms with an unaltered value for the 
ratio of its sides. Specially, when O passes to infinity on x, and 
the circle becomes a line normal to a, inversion becomes reflexion 
in this normal and produces an image of the original figure. 
It will be convenient to suppose that the arbitrary size of the 
figure is kept always such that the product AA’. BB’. CC’. DD’ 
remains constant. This is secured by making F‘ proportional to 
OA.OB.OC.OD. An equivalent form of the constant is 
(AB? — AB?) (CD? — CD?), 
and hence AB.CD is constant and also AB’.CD, AB.C’D and 
AB’.C’D. The product of any two sides taken one from each of : 
a companion pair of isograms remains constant. 
§ 3. From the figure of § 2, which may be called a symmetro- 
gram, may now be constructed the isogram double-four. The 
geometry is best expressed in terms of vector multiplication; the 
product of two vectors being equal to the product of two others 
if the product of their lengths and the sum of their angles are in 
each case the same 
