16 IT. S. COAST AND GEODETIC SURVEY. 



With these values we obtain 



ds a(l — e^) • a cosec^ <p _ d ^ cos^ <p 



dip (1 - €2 sin 2 ^)^/2 (1 - e^ sin2 ^)i/,'^ (i _ ,2 ^^^2 ^yi. 



a(l — cosec^ v?) _ a cot^ <^ 



Xl-e2sin2<^)V2~ ~(l-e2sin2,^y/2' 



hence 

 Therefore 



1 ds 



- J- = — cot (p. 

 p Clip 



1 dv, 



\ip 



— J— — —cot (p: 

 udc ^' 



by integration, we obtain 



log u= —log sin <;c' = log cosec (p, 

 or, passing to exponentials, 



'z^ = cosec (p. 



But 



^ 6 r(x) ^,^, . 



tan ^ =^^^ =r(X) sm <^. 



The length of an arc of the developed parallel is given by 



a a 



2a cot (p fl 2 ^^ ^^^ ^ 2 



p0== tan 2 0= r(X) ^. 



(1 — e^ sin^ (pyi^ tan ^ (1 — e^ sin^ (pfl^ tan :^ 



On the equator, since ip = and ^ = 0, we obtain for an arc 

 from X = to X the value 



equatorial arc = 2a r(X). 



If we now add the condition that the equatorial arcs are 

 to be preserved in their true length, we have 



2ar(X) = aX 

 or 



r(x)=|. 



