124 U. S. COAST AND GEODETIC SURVEY. 



The two equations are 



sin (p=JlS — tan^ ^ jtan ^ 



X = ^^ 3 + tan^ Y )tan "2 • 

 Tcja and Icp have now become 



X. _ V^ cos <p (1 +COS <p'y 



^~ ^ COS <p^ (1 +COS X' COS <^0 



X. __i_ COS (p^ (1 +COS yy 



^ -y/ir COS (p (1 +COS X' COS ^0 



zr-i. I. _ri (l+cosXQ (l+cos<^0 "|^ 



it-/^ni%-[^2 1+C0SX'C0S<P' J* 



The latter formula can be written 



j^r. _1 (l-cosXQ (l-cos^O n^ 

 L 2 1+cosX' cos<p' J * 



In this form we see that K is everywhere less than unity, 

 except on the Equator and upon the central meridian, and 

 that the alteration of surface increases with the longitude 

 and with the latitude. On the principal meridian we 

 obtain 



Z = cos*|-. 



Let us further examine how <p^ ought to vary with <p in 

 order that the areas should be preserved along the prin- 

 cipal meridian. If we denote by ?i" the value which the 



derivative of X' with respect to X takes for X = -^, we should 

 have 



cos (p d(p = n'' R^ cos (p' dip^ 

 or, by integration, 



sin ip = n" B? sin <p', 



no constant being added, since (p and <^' vanish simul- 

 taneously. 



