114 



IT. S. COAST AXD GEODETIC SURVEY. 



Hence ^0=0 ^^^ ^^^ Equator is represented by a straight 



line. The whole surface of the earth may be represented 

 on a unit circle with the projection as defined, and the 

 projection is so given in figure 34. 



^G. 34.— Lagrange's projection, earth's surface in a circle. 



EQUIVALENT OR EQUAL-AREA POLYCONIC PROJECTIONS. 



An equivalent or equal- area projection is one in which 

 the proportion of areas is preserved constant; that is to 

 say, that any portion of the map bears the same ratio to 

 the region it represents that any other portion does to the 

 region which it represents, or the ratio of area of any part 

 is equal to the ratio of area of the whole representation. 

 This is expressed analytically by the equation 



^m^pCOS^=l. 



In the poly conic projection this becomes for the sphere 



p /ds - dp\dd ^ 

 -^ — ( 3- cos - j^ ) 5^ = 1 . 

 a^cos (p\a<p dip J Oh 



