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].