1. This expression is easily derived by a complex-variable scheme for handling gradients of two-dimensional wire systems. If z = x+iy is a complex number representing any point in the plane of Fig. 2 (not to be confused with thez-coordinate of Fig. 1), and Z = X+iY represents the coordinates of one of the wires, then the gradients created when this wire carries currentI into the papercan be represented by the complex series: ∂Hy∂x+i∂Hx∂x = 0.2 I(Z−z)2 = 0.2 IZ21+2zZ+3z2Z2+4z3Z3+⋯. This series can be easily summed over theZ-values of the eight wires of Fig. 2 to obtain the series in (2). To the same accuracy as the longitudinal gradient ∂Hx∕∂x is constant, the transverse gradient ∂Hy∕∂x is found to vanish: ∂Hy∂x = 83I10R29r8R8sin 8φ−13r12R12sin 12φ+⋯.