146 U. 8. COAST AND GEODETIC SURVEY. 



By substituting these values in the differential formulas 

 B obtain 



we 



a oot ^ a cot (p . ^ 



jz 5 — 7—= — TjT- A cos (f — 73 5 — =— ^ — xi2 sm d 



. (1 — e^ sin^ <p) /' ^ (1— €^sm^^)^ 



^~ g cof^ ip a cosec ^ip — ae^ (1 + cos^ 



(1 - e^ sin^ ^)>^ ^^® ^ + (1 - e^ sin^ ^)'/^ 



. - X cos^ (^ sin <p — cos^ </> sin 6 



tan ^ = :r-^ TT^r-, 5 \^ — ^"T" 



2 ^ , 1 e^ (1 +cos2 <p) sm^' <p 



— COS^ (p cos + ^i 5—^-^ — ^^j 5-^^-1 ^ 



^ 1 — e^ sin^ <^ 1 — e^ sin^ (p 



X sin (p — sm 6 



~ , /I €^ sin^ (p 



sec^ ^— cos B—z o • o 



^ 1 — e^ sm^ (p 



^-sin ^ 



„ - 6^ sm^ <p 

 sec^ ^ — cos 6—-^ , . 2 



I. (1 — e^ sin^ ^)''/' r a cot^ ^ a cosec'^ y? 



'^"^ ~ a (1 - e^) |_ (1 _ ,2 sin2 ^)>^ cos ^ + ^ _ ^2 g^^a ^)ji 



ae^ cos^ ^ "I , 



sec 1^ 



= .. _ 2 [ — (1 — €^ sin^ <^) cot^ <p cos ^ + cosec^ ip (1 — e^ sin^ tp) 



— ^ cos^ <^] 



sec \^ r 

 = .. __ 2 cosec^ ^ — e^ — e^ cos^ <p — cot^ <^(1 — €^ sin^ ip) 



(1-2 sin-*)] 

 = 1 _ 2 cosec^ ip — ^ — ^ cos^ ^ — cot^ ^ + e^ cos^ ^ 



+ 2 (cot^ ^ — €^ cos^ ip) sin^ ^ 

 = 1^ [1 - e^ + 2 (cot^ ^ - e^ cos^ <^) sin^ |1 



2 (cot^ ^ — e^ cos^ ^) sin^ -h 



K=\+ ^-^ 2, 



l-e^ 



