90 U. S. COAST AND GEODETIC SURVEY. 



But for the sphere 

 or for the spheroid 



Therefore, for the sphere 



2cZ: sinnX tan^ (l'^^) 



a;2 + 



x = 



y= 



c 



y — 



c[^^tan-(|+|)-l] 



[p tan^^j+|)+ l]r 4c^ P tan - (|+|) 

 "^[Ptan-(|+|)-l] 



Ptan^^| + f)-l 



We thus see that 



Ptan-(f+|)+l 



= c 



2c^ tan'' 



(M) • 



Jt2 tan^^ 



(M)- 



If we denote that intersection which lies nearest the origin 

 by yo (that is to say the y value for X = 0) , we have 



tan- = ^^^ = ^"^"^^ 



