P[beta] being the value of 2p sin z when z = [beta]. This gives d²y dy /dy\ sin² z --- + sin z cos z -- - y = z - sin z ( -- )[beta] = 0. dz² dz \dz/ This method of development is due to Sir George Airy, whose original paper--the investigation is different in form from the above, which is due to Colonel Clarke--will be found in the _Philosophical Magazine_ for 1861. The solution of the differential equation leads to this result-- [rho] = 2 cot ½z log_e sec ½z + C tan ½z, C = 2 cot² ½[beta] log_e sec ½[beta]. The limiting radius of the map is R = 2C tan ½[beta]. In this system, called by Sir George Airy _Projection by balance of errors_, the total misrepresentation is an absolute minimum. For short it may be called _Airy's Projection_. Returning to the general case where [rho] is any function of z, let us consider the local misrepresentation of direction. Take any indefinitely small line, length = i, making an angle [alpha] with the meridian in co-latitude z. Its projections on a meridian and parallel are i cos [alpha], i sin [alpha], which in the map are represented by i[sigma] cos [alpha], i[sigma]´ sin [alpha]. If then [alpha]´ be the angle in the map corresponding to [alpha], tan [alpha]´ = ([sigma]´/[sigma]) tan [alpha]. Put [sigma]´/[sigma] = [rho]dz/sin z d[rho] = [Sigma], and the error [alpha]´ - [alpha] of representation = [epsilon], then ([Sigma] - 1) tan [alpha] tan [epsilon] = -------------------------. 1 + [Sigma] tan² [alpha] Put [Sigma] = cot²[zeta], then [epsilon] is a maximum when [alpha] = [zeta], and the corresponding value of [epsilon] is [epsilon] = ½[pi] - 2[zeta]. For simplicity of explanation we have supposed this method of development so applied as to have the pole in the centre. There is, however, no necessity for this, and any point on the surface of the sphere may be taken as the centre. All that is necessary is to calculate by spherical trigonometry the azimuth and distance, with reference to the assumed centre, of all the points of intersection of meridians and parallels within the space which is to be represented in a plane. Then the azimuth is represented unaltered, and any spherical distance z is represented by [rho]. Thus we get all the points of intersection transferred to the representation, and it remains merely to draw continuous lines through these points, which lines will be the meridians and parallels in the representation.