92 TJ. S. COAST AND GEODETIC SURVEY. 



The equation for the parallel becomes 



2 r c(m^— l) l^_ 4c^ m^ 



The equation of the meridians remains as before 



(x + c cot n\)^ + y^ = c'^ cosec^ n\. 

 The coordinates expressed in terms of m become 



2cm sin nk 



x = 



1 + 2m cos n\ + m^ 



c (m^— 1) 



^ ~ 1 + 2m cos rtX + m^' 



and tne magnification for the sphere becomes 



7, _ 2cm7i 



~ a cos (p (l+2m cos n\ + m^y 



and for the spheroid 



, , _ 2cmn -yjl — e^ sin^<^ 



~ a cos tp (l + 2m cos n\-{-w?) 



with the value for m in the last form 



\4 2/ Vl+€sm^/ 



Since both ^ and a must be less than ^, if (p is greater than 

 — a, then 



tanQ+|)>tan(j-0 



or 



*^'*(¥+2)**'^(l+i)>^ 



and 



m>l. 



In a similar way it may be shown that when ^< — a, then 

 m<l. 



