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Full text of "Scientific Papers - Vi"

373.
ON  CONFORMAL REPRESENTATION FROM A MECHANICAL
POINT OF VIEW.
[Philosophical Magazine, Vol. xxv. pp. 698  702, 1913.]
IN what is called conformal representation the coordinates of one point x, y in a plane are connected with those of the corresponding point , t] by the relation
>), .............................. (1)
where / denotes an arbitrary function. In this transformation angles remain unaltered., and corresponding infinitesimal figures are similar, though not in general similarly situated. If we attribute to f, 77 values in arithmetical progression with the same small common difference, the simple square network is represented by two sets of curves crossing one another at right angles so as to form what are ultimately squares when the original common difference is made small enough. For example, as a special case of (1), if
x + iy = c sin ( + i^),  ........................... (2)
as = c sin f cosh 77,   y = c cos  sinh 77 ; and the curves corresponding to 77 = constant are
.(3)
.(4)
+      .".....,,,T-,,-,  i
c2 cosh2 77    c2 sinh2 77
and those corresponding to  = constant are
n&                      nfi.
_________y_____ ]
c2 sin21    c2 cos2 f      '
a set of confocal ellipses and hyperbolas.
It is usual to refer ae, y and , 77 to separate planes and, as far as I have seen, no transition from the one position to the other is contemplated. But of course there is nothing to forbid the two sets of coordinates being taken in the same plane and measured on the same axes. We may thenortion of this paper.                      lii ----  /(< ---   7    Tr~77 7 \  i         7        .