Elliptic Integrals - Map Projections: Cartographic Information Systems - page 696

Proof (Proof of formulae ( C.10 )and( C.11 )).

y := cn u = √ 1 x 2 , sn u = 1

y 2 = x,
dn u 1 − k 2 (1 − y 2 )= √ 1 − k 2 x 2 ,

− −

(C.22)

d u = d √ 1

d y − x 2 x d x

d u ,

= − √ 1

d u − x 2

x √ 1

d y
d u =

− − k 2 x 2 = − sn u dn u q . e . d . (C.23)

d y
d u

2

y 2 )(1 k 2 + k 2 y 2 )q . e . d .

=(1 − −

End of Proof (Proof of formulae ( C.10 )and( C.11 )).

Proof (Proof of formulae ( C.12 )and( C.13 )).

k 2 x 2 , sn u = √ 1

z := dn u = √ 1

− z 2

− = x,

k

cn u = k 2

= √ 1

− (1 − z 2 )
k

− x 2 , (C.24)

d u = d √ 1

k 2 x

d z − k 2 x 2 d x

d u ,

√ 1

= −

d u

− k 2 x 2

d z

d u = −k 2 sn u cn u q . e . d .

(C.25)

k 2 x √ 1

√ 1

z 2 k 2

d z
d u =

− − x 2 = − − − (1 − z 2 ) ,

d z
d u

2

=(1 − z 2 )[ z 2 − (1 − k 2 )] q . e . d .

End of Proof (Proof of formulae ( C.12 )and( C.13 )).

Proof (Proof of formula ( C.15 )).

x =sn u =cn u dn u

⇒