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Full text of "Scientific Papers - Vi"

154                         ON CONFORMAL REPRESENTATION FROM A                         [c
regard the angular points of the network as moving from the one posit to the other.
Some fifteen or twenty years ago I had a model made for me illustrat of these relations. The curves have their material embodiment in wires hard steel. At the angular points the wires traverse small and rather th brass disks, bored suitably so as to impose the required perpendicularity, 
Fig. l.
two sets of- wires being as nearly as maybe in the same piano. But soi thing more is required in order to secure that the rectangular elemeni the network shall be square. To this end a third set of wires (shown dot in fig. 1) was introduced, traversing the comer pieces through bori making 45 with the previous ones. The model answered its purpose t certain extent, but the manipulation was not convenient on account of friction entailed as the wires slip through the closely-fitting corner pie Possibly with the aid of rollers an improved construction might be arrived
The material existence of the corner pieces in the model suggests consideration of a continuous two-dimensional medium, say a lamina, wh deformation shall represent the transformation. The lamina must be such a character as absolutely to preclude shearing. On the other hand must admit of expansion and contraction equal in all (two-dim enaioi directions, and-if t-he-deformation is to persist without the aid of appl forces, such expansion must be unresisted.
Since the deformation is now regarded as taking place continuously, J (1) must be supposed to be a function of the time t as well as of  4- wj. may write
<K + iy=f(t> % + iri)............................(5
The component velocities u, v of the particle which at time t occupies position x, y are given by dx/dt, dy/dt, so that
(6  J-                 / /?  7\       -MW;