2 
In order to take into account the flattening of the pole, i.e. to preserve, 
as far as possible, correct meridional distances between latitudes, and to 
eliminate extreme errors on maps of moderately small scales, a process 
may be evolved on the same hypothesis in the assumption of a series of 
spheres whose radii would gradually increase with the latitudes, so that 
the least errors would be found in the lengths of corresponding parallels. 
Such a conventional projection would be the result of a superposition or 
assemblage of separate projections of zones of different spheres, closely 
approximating natural surfaces comprised between common axes of rect- 
angular co-ordinates and the parallels of latitude, whereby true distances 
would be held, at least, on the central meridian. 
The meridian of longitude 95 degrees west from Greenwich, is chosen 
as the central meridian — or auxiliary equator — and the intersection of the 
parallel of latitude 60 degrees with this meridian, as the origin of rectangular 
co-ordinates. The unit of distances for the computation is the statute mile. 
DETERMINATION OF THE RADII OF THE AUXILIARY SPHERES 
The radii of the successive spherical zones are determined from the 
true lengths of meridional arcs measured from the origin of co-ordinates 
at latitude 60 degrees to the successive intersections of parallels of latitude. 
These meridional distances may be calculated by the standard formula in 
geodesy, but a more expeditious method is to compute them from tables 
given in Special Publication No. 5 of the U.S.C. and G. Survey, thus: 
Denoting by mi m3. . . .mn, the lengths of single degrees of latitude 
north of latitude 60 degrees ; by m_i m_2 m_3 .... m_n, those south of latitude 
60 degrees as found in the tables, and hy Mi M 2 .... etc., the meridional 
distances, we have: 
Between lat. 60° and lat. 61°, (1 degree): Mi = mi 
“ “ 62°, (2 degrees) : ikfa = mi + m2 
“ “ 63°, (3 degrees): Mz = mi -h m2 + m3 
“ “ (60 -(- n)°, (n degrees): M^ = mi m 2 + mz 
+ + Wn 
and likewise; 
Between lat. 60° and lat. 59°, (1 degree) : M.i = m_i 
** 58°, (2 degrees) : ilf_2 = m_i + m_2 
** 57°, (3 degrees): M .3 = m.i + m_2 + w-3 
** (60 — n)°, (n degrees); M^a = w_i + m_2 + w_3 
H" + m.n 
Denoting by ri ra ra and r_i r_2 r.j . . . . r_n, the radii of spheres 
corresponding to those meridional arcs, we have 
ri = Ml (360 /2ir) 
M 2 (360 /2x) 
ra = 
2 
rz 
rn 
r.i 
Mz (360 /2x) 
3 
M^ (360 /2x) 
n 
M^i (360 /2x) 
