474 21 Datum problems
Continuation of Box.
s 26 = 0 ,
s 20 =
= − cos b sin b(qE
2 − e
2 )n/(qm + h) =
= − cos b sin b(qE
2 − e
2 )n/qm + h cos b sin b(qE
2 − e
2 )n/(qm)
2 + O(h
2 ) ,
s 21 =
= sin b cos l/(qm + h) =
= sin b cos l/(qm) − h sin b cos l/(qm)
2 + O(h
2 ) ,
s 22 =
= sin b sin l/(qm + h) =
= sin b sin l/(qm) − h sin b sin l/(qm)
2 + O(h
2 ) ,
s 23 =
= − cos b/(qm + h) =
= − cos b/(qm) + h cos b/(qm)
2 + O(h
2 ) ,
s 24 =
= + sin l(a
2
1 + hn)/[n(qm + h)] =
= +a
2
1 sin l/(nqm) + h sin l(nqm − a
2
1 )/(nq
2 m
2 ) + O(h
2 ) ,
s 25 =
= − cos l(a
2
1 + hn)/[n(qm + h)] =
= −a
2
1 cos l/(nqm) − h cos l(nqm − a
2
1 )/(nq
2 m
2 ) + O(h
2 ) ,
s 27 =
= e
2 sin b cos bn/(qm + h) =
= e
2 sin b cos bn/(qm) − he
2 sin b cos bn/(qm)
2 + O(h
2 ) .
(21.69)
As soon as we implement the curvilinear datum transformation close to the identity (21.67) into
the global representation (21.66) of conformal coordinates, we are led to the first version of global
conformal coordinates (21.70) of Box 21.14 in terms of local ellipsoidal coordinates: see Box 21.15
and Box 21.16. Note that the coefficients {U MN , V MN } in Box 21.15 and Box 21.16 depend on the
coefficients {X MN , Y MN } which, in turn, are given in terms of the parameters A 1 , E
2 , B 0 . Accordingly,
all the coefficients are transformed under the change of the form parameter E
2 from the global system
of reference to the local one. The evaluation points {L 0 , B 0 } and {l 0 , b 0 } have been chosen to be
identical in order to conserve the meridians-of-reference in both reference systems as well as the
parallel-of-reference. Box 21.17 is a collection of Taylor expansions of order one of conformal series
coefficients under a variation of form parameter δE
2 listed separately in Box 21.18 and Box 21.19.
In consequence, upon a replacement of global coefficients by local coefficients of Box 21.18 in the
coefficients of Box 21.19 and Box 21.15 , we derive the second version of global conformal coordinates
(21.70) in terms of local ellipsoidal coordinates with the coefficients of Box 21.20 and Box 21.21.
Continuation of Box.
s 26 = 0 ,
s 20 =
= − cos b sin b(qE
2 − e
2 )n/(qm + h) =
= − cos b sin b(qE
2 − e
2 )n/qm + h cos b sin b(qE
2 − e
2 )n/(qm)
2 + O(h
2 ) ,
s 21 =
= sin b cos l/(qm + h) =
= sin b cos l/(qm) − h sin b cos l/(qm)
2 + O(h
2 ) ,
s 22 =
= sin b sin l/(qm + h) =
= sin b sin l/(qm) − h sin b sin l/(qm)
2 + O(h
2 ) ,
s 23 =
= − cos b/(qm + h) =
= − cos b/(qm) + h cos b/(qm)
2 + O(h
2 ) ,
s 24 =
= + sin l(a
2
1 + hn)/[n(qm + h)] =
= +a
2
1 sin l/(nqm) + h sin l(nqm − a
2
1 )/(nq
2 m
2 ) + O(h
2 ) ,
s 25 =
= − cos l(a
2
1 + hn)/[n(qm + h)] =
= −a
2
1 cos l/(nqm) − h cos l(nqm − a
2
1 )/(nq
2 m
2 ) + O(h
2 ) ,
s 27 =
= e
2 sin b cos bn/(qm + h) =
= e
2 sin b cos bn/(qm) − he
2 sin b cos bn/(qm)
2 + O(h
2 ) .
(21.69)
As soon as we implement the curvilinear datum transformation close to the identity (21.67) into
the global representation (21.66) of conformal coordinates, we are led to the first version of global
conformal coordinates (21.70) of Box 21.14 in terms of local ellipsoidal coordinates: see Box 21.15
and Box 21.16. Note that the coefficients {U MN , V MN } in Box 21.15 and Box 21.16 depend on the
coefficients {X MN , Y MN } which, in turn, are given in terms of the parameters A 1 , E
2 , B 0 . Accordingly,
all the coefficients are transformed under the change of the form parameter E
2 from the global system
of reference to the local one. The evaluation points {L 0 , B 0 } and {l 0 , b 0 } have been chosen to be
identical in order to conserve the meridians-of-reference in both reference systems as well as the
parallel-of-reference. Box 21.17 is a collection of Taylor expansions of order one of conformal series
coefficients under a variation of form parameter δE
2 listed separately in Box 21.18 and Box 21.19.
In consequence, upon a replacement of global coefficients by local coefficients of Box 21.18 in the
coefficients of Box 21.19 and Box 21.15 , we derive the second version of global conformal coordinates
(21.70) in terms of local ellipsoidal coordinates with the coefficients of Box 21.20 and Box 21.21.