406 
DESIDERATA AND SUGGESTIONS. 
[694 
A, B, G,..., and the values £, f l5 £ 2 ,... by points P, P 1} P 2 ,... the first of which is 
assumed at pleasure, and the others each from the preceding one by the like given 
geometrical construction. The problem is to determine the regions of the plane such 
that, P being taken at pleasure anywhere within one region, we arrive ultimately at 
the point A ; anywhere within another region at the point B; and so for the several 
points representing the roots of the equation. 
The solution is easy and elegant in the case of a quadric equation: but the next 
succeeding case of the cubic equation appears to present considerable difficulty. 
Cambridge, March 3rd, 1879. 
No. 4. The mechanical construction of conformable figures. 
[From the American Journal of Mathematics, t. II. (1879), p. 186.] 
Is it possible to devise an apparatus for the mechanical construction of conformable 
figures; that is, figures which are similar as regards corresponding infinitesimal areas ? 
The problem is to connect mechanically two points P 1 and P 2 in such wise that Pj 
(1) shall have two degrees of freedom (or be capable of moving over a plane area) 
its position always determining that of P 2 : (2) that if P 1} P 2 describe the infinitesimal 
lengths PiQj, P 2 Q 2 , then the ratio of these lengths, and their mutual inclination, shall 
depend upon the position of P lt but be independent of the direction of PjQi: or 
what is the same thing, that if P 2 describe uniformly an indefinitely small circle, 
then P 2 shall also describe uniformly an indefinitely small circle, the ratio of the 
radii, and the relative position of the starting points in the two circles respectively, 
depending on the position of Pj. 
Of course a pentagraph is a solution, but the two figures are in this case 
similar; and this is not what is wanted. Any unadjustable apparatus would give one 
solution only: the complete solution would be by an apparatus containing, suppose, a 
flexible lamina, so that P l moving in a given right line, the path of P 2 could be 
made to be any given curve whatever. 
Cambridge, Jidy 9th, 1879.