Companion to Concrete Mathematics, Vol. 1 by Zdzislaw Alexander Melzak

By Zdzislaw Alexander Melzak

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By the foregoing, N is therefore Nx-l y- 1 - Nx-l y+ 1 which gives for the desired probability N Nxy p-q p +q Related use of reflections occurs in the solution of certain problems in electrostatics by Kelvin's method of images. In the simpler cases the object is to calculate the potential due to a point charge e in the presence of metal surfaces kept at 0 potential. The correct boundary condition, namely the 0 potential, may be sometimes synthesized by placing a finite or infinite number of charges ±e at the images, or repeated images, of the original point charge in the loci of the 0 potential.

There is some reason to believe that it is not, and the reader may try to relax that property. Whether our solution holds under the modest assumption of mere existence of the maximizing arc C1 , appears to be still unknown. We mention briefly the isodiametric problem of the circle: of all plane sets, no two points of which are further than one unit apart, to find the set X whose area A(X) is largest. We suppose that such a maximizing set X exists and, as in the isoperimetric problem, we show easily that X is convex.

We apply first the principle of minimum perturbation and we fix X and Y while Z varies on AB. XZ and YZ being variable, the sum XZ + YZ is to be minimized; hence by the previous problem we find 1:: XZA = 1:: YZB. Similarly, by varying Y alone and then X alone we find that -i:CYX = -i:Z YB and -i:AXZ = 1:: CXY. Now reflect Z in AC into Z 1 and in BC into Z 2 • By the laws of reflection Z 1 X = ZX and Z 2 Y = Z Y; by the foregoing angle-equalities the points Z 1 , X, Y, Z 2 are collinear. Heilce the perimeter of XYZ is Z1 Z 2 • Further, Z1 C = ZC = Z 2 C and 1::Z1 CZ 2 = 2-i:ACB.

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