4 
Let A be a point on the earth’s surface whose latitude and difference 
of longitude with the central meridian are respectively ^ and X. Through 
point A, draw an arc of great circle P' A B perpendicular to the central 
meridian, and an arc of small circle A A', parallel to the central meridian. 
The arc of great circle DAP perpendicular to the equator represents the 
meridian of point A. 
In the right-angle spherical triangle A P, formed by those arcs of 
great circle, we have the following relations: 
sin u = sin (90° — <i>) sin X 
= cos ip sin X (2) 
^ cos X 
tan V = 
cot (90° — <f>) 
~ cos X tan (90° — <f>) 
= cos X cot ^ (3) 
Now, in a polyconic projection of various points, A, laid in a transverse 
position, the pole P being transferred to P' on the equator, and the apexes 
of the tangent cones lying on the axis of abscissae, the values (2) and (3) 
may be assumed, for purpose of computation, to be the auxiliary latitude 
and longitude respectively as elements of the formulae of the regular 
polyconic development. Thus, we have from Figures 1 and 2; 
r = radius of spherical zone 
I = r cot u 
{ a = (30° — v) sin u (where v < 30°) 
\ a = (y — 30°) sin u (where v > 30°) 
u' = 21 sin 2 - 
2 
or u* = 2r cot u sin^ — 
2 
The length of arc u in statute miles is given by the formula 
Wm = 2 TT r 
u 
360 
