THEORY or POLYCONIC PROJECTIONS. 147 



When X is small — that is, when the map is not extended 

 far from the central meridian — an approximation in a 

 series in terms of X is very convenient. If we neglect ^ 

 and higher powers, we obtain 





^ 1 — e^ sm^ <p 2 



X^ sin^ (p 



tan ^= 



^ , e^ sin^ (p , X^ sin^ (p 



tan^ (p—- ' 



l-c^sm^^ 2 



or approximately 



X^ sin^ <p.^ o • , V 

 g — ^(l-e^sm^ <p) 



tan \l/ = 



tan^ (p(l — €^ sin^ <p) — e^ sin^ <p 



X^ . 2 /l-e^sin^v'X 



= 6 sm ^ cos^ <P {^ — iZTeT-) 



= Y2SJ^2(pcos^( — Yzr^j- 



For smaller values of ^ this can be stUl further approxi- 

 mated by the form 



^= Y2 ^^ ^^ ^^® ^'^ 

 for the sphere km becomes 



km = sec ^ (cosec^ ^ — cot^^ cos $). 

 To obtain an approximation we let sec ^=1 and we get 



km=( cosec^ <p — cot^ <p+cot\(p 2-— • • • •) 



X^ 



= l+2'COs^^. 



