126 TJ. S. COAST AND GEODETIC SURVEY. 



The ratio of surfaces is greater than unity everywhere 

 except on the Equator and upon the principal meridian. 

 The alteration increases with the latitude; on the other 

 hand, it diminishes when the longitude increases. This is 

 shown at once by writing the above expression in the form 



K= sec^ 



\ 1+ cos ip COS X V 



Upon the central meridian, where the greatest alteration is 

 produced, we have 



Z=sec*|. 



The conditions to insure that the areas should be pre- 

 served along the meridian of longitude \ and along the 

 parallel of latitude <po gi^e, respectively, the differential 

 equations 



. . , cos ip' 1 ^^' =1 



^ ™ ^ « cos <p (1 +COS X'o COS ^y d<p 



_ . , , 1 dy_^ 



Bsm<Po tan <P o ^^_^^^^ ^^^ ^^^yy-^-^- 

 The integration of the first- equation gives 



and from the second we get 



. T>r 4 , lA /o^ X'\ sin X^ 1 



\ = B\ -. — FT-r tan-M tan ^ tan -^ ) — i— ; / ^^^\f * 



Lsm 2(p\ \ 2 2 / 1 + cos v? cos X J 



The quantities (fo, (p\, Xq, X'o and the constants A and 5 are 

 joined to each other by the four relations that are obtained 

 by expressing that the first equation is satisfied for <^ = ^o 



with (p' = ^p'q, as also for (p = 2 ^^^ ^' ^2 ^^^<i the second for 

 X = 2 with X' = 2, as also for X = Xo with X'=X'o- 

 The ratio of surfaces has now become 



^_ r (1 + cos X'o cos <p') (1 + COS <p'o COS X0 "| 2 ^ 

 ~L(l+COS9'oCOSX'o) (1+cos X' cos<^Oj * 



