THEORY OF POLYCONIC PROJECTIONS. 149 



customary, however, in the cons true tiou of the projection 

 to locate the various origins on the central meridian by 

 their meridional-arc values and then to use the coordinates 

 as originally computed. It is, in general, not necessary to 

 compute the Pn values since the tabulated A factor values 

 given in Special Publication No. 8, United States Coast 

 and Geodetic Survey, are connected with them by the 

 relation 



A 



Pnsin \" 

 or 



1 

 ^^~^sinl"* 

 Hence 



log Pn = colog A + colog sin 1 ". 



The logarithms of the A factors in meters are tabulated for 

 each minute of latitude in Special Publication No. 8, as 

 referred to above. With these values as given the formula 

 for p becomes 



P = P^ cot <p, 



A great advantage of this projection consists in the fact 

 that a universal tm)le can be computed that can be used 

 anywhere upon the earth's surface. Almost every other 

 projection has special elements that must be determined 

 lor each projection. These elements are generally certain 

 arbitrary constants that enter into the formulas for compu- 

 tation. The Mercator projection is another projection that 

 can have a universal table. 



If the whole earth's surface were mapped in one continu- 

 ous projection it would be interesting to know what would 

 be tne length of the meridian that forms the outer boimdary 

 of the representation and also how many times the area has 

 been increased. Such a projection of the sphere is shown 

 in figure 40. By approximate measurement on a plate of 

 such a projection it was found that the ratio of increase of 

 length of the outer meridian was about 3.2 to 1. 



The element of area of the representation beiog given in 

 the form 



dS = a? K co^ <p dip dk 

 for the sphere, we have 



K= (cosec^ <p — cot^ <p cos d), 



