long geodesics on the ellipsoid - hf rainsford

LONG GEODESICS ON THE ELLIPSOID

H. F. RAINSFORD

Senior Computer, Directorate of Colonial Surveys, Tolworth (G.B.)

There are two main problems concerning long lines:

(a) Given the geographical co-ordinates of two points, to find the distance and azimuths between them; commonly called the reverse (or inverse) problem;

(b) Given the geographical co-ordinates of one point and the distance and azimuth from it to another, to find the co-ordinates of the second point; commonly called the direct problem.

The leaders in making a general study of the geodesic were EVLER, LEGENDRE and BESSEL. The two main problems were solved by relating the geodesic elements to the corresponding elements of an auxiliary spherical triangle in which the reduced latitudes were used instead of the geographic latitudes. The same method was given by JORDAN, HELMERT and CLARKE. It is a solution in terms of trigonometrical functions. However, successive approximation is necessary to find the longitude angle of the auxiliary spherical triangle used. This is a tedious process (particularly with 10- figure tables) as at least three approximations are required for a precise result. The two problems should be considered together as each can be used to check the other by different formulae. Recently LEVALLOIS and DuPuY have proposed a slightly different solution using tables of 'Wallis Integrals' o r f sinZPx dx. The main advantages are that the variation in tabular factors required is comparatively slow and the series converge more rapidly. The tables are now available only for centesimal arguments. It is probable that this method would be widely used if the tables were given in sexagesimal arguments and for several Figures of the Earth in general use. The methods given in this article can be used to obtain absolute precision for any length of line, without special tables (except possibly those for the principal radii of curvature of the spheroid).

2. Formulae for long lines should be applicable to any length of line up to halfway round the world. To clarify one's ideas on comparable accuracy in length, azimuth and position, suppose that an accuracy of 1 mm is required and take the major axis of the spheroid (0.638 107 metres) as the radius of a spherical earth. Then i mm on the surface of the earth is equal to 0".00003 in position: also, a variation in azimuth of 0".00001 corresponds to a lateral shift of 1 mm at a distance equal to half round the world. Since both problems require the solution of auxiliary spherical triangles, it would appear that 10-figure tables (working to the fifth decimal of a second) are necessary to obtain accuracy to the millimetre. Such accuracy may be required for the very precise checking of reverse and direct computations; in particular, when the examples concerned are to be

12


used for comparison with results by more approximate methods. Accuracy to a centimetre (approximately) may be obtained with 8-figure trigonometric tables provided that 9- or 10-figure computat ion is used when required for the first term of any series.

o

/

P

~0 a~

Figure 1

3. Figure 1 i l lustrates the auxil iary spherical tr iangle used in the reverse problem. AEFG is the equator and P the pole. B and C are the two points on the sphere corresponding to the two given spheroidal points. The great circle through B and C cuts the equator at A and has its point of max imum reduced lat itude at D. BE and CF are the reduced latitudes U 1 and Uz corresponding to the latitudes ~Pl and q~2- The spherical azimuths % and a2 are precisely the same as the corresponding spheroidal azimuths. L is the difference of longitude on the spheroid and X the corresponding difference on the sphere. The azimuth of the great circle AD at the equator is ~. The arcs (Yl (AB) and a2 (AC) are measured from the equator. The arc BC=a2-a l =~ and 2~=at +a2. I f a and b are the major and minor axes of the spheroid, other spheroidal parameters a re : - -

e 2 ___ (a 2.b2)/a2

q2 = (a2_b2)/b2

f = (a-b)/a

4. Since X is unknown at first, the spherical tr iangle is solved using >, =L as a first approximat ion to find the correction (X -L ) . The process is con- t inued until further repetit ion will not alter the results: usually, three approximations are sufficient. The most convenient formulae for machine solution are :

b b . . . . tan qo 2 tan U l -a tan ~01; tan U2- a

cosa=s in U 1 sin U 2+ cos U 1 cos U 2cosx

sin sc = cos U 1 cos U 2 sin X/sin

sin 0q cos U 1 =sin ~2 cos U 2 =sin oc

cos 2a~ = cos a -2 sin U 1 sin U2/cos 2 s~

. . . . (1 )

. . . . (2 )

. . . . (3 )

. . . . (4 )

. . . . (5 )

13

NOTICES SCIENTIFIQUES

cos 4am = 2 cos 2 2era - 1

cos 6~m =4 cos 3 2~m-3 COS 2~m

sin 2~ = 2 sin a cos

sin 3~ = 3 sin ~ -4 sin 3

fAocr +A 2 sin cr cos 2cry, (X-L) =fs in x L +A 4 sin 2 . cos 4*m +A6 sin 3or cos 6cr,.

. . . . (6 )

. . . . (7 )

. . . . (8)

. . . . (9 )

. . . . (10)

A 0 = 1 - ~f (I +f+f2) cos 2 ~ + 1.~f2 (1 + 9 f ) cos 4 a _ l_~sf3 cos 6 a]

A2=88 +S+f2) cos 2 ~ _if2< (1 +~f) cos4 ~ + ~-F6f3 cos6 ~ A 4 =~2-f 2 (I +9f ) cos 4 oc- ~-!3~f3 cos 6 o~ [ . . . . (11)

A 6 =7~8f 3 cos 6 ~ J

The A coefficients are given as functions o f f since they converge more rapidly than when given as functions of e2. The maximum value of any term in f4 (i.e.f3 in the A's) is less than 0".00001 even for a line half round the world. Thus the A 6 term may be omitted altogether and the following simplified forms used even for precise results:

A 0 = 1 - 88 +f ) COS 2 ~ + 1-~f 2 COS 4 0r 7 I

A 2 = i f (1 +f ) COS 2 0~ __ 88 COS 4 ~ !

A 4 =)-lz-f2 cos 4 o~ J

(12)

5. To avoid the successive approximation a method has been produced by the Army Map Service (U.S.A.)I which gives the correction (X-L ) directly by solution of the triangle BCP in which the angle at P is taken as L (the spheroidal longitude).

Let x = sin U 1 sin U2 "] . . . . (13)

y = cos U1 cos Uz A Let 0 be the first approximation to cr and z the first approximation to

sin ~, then

cos 0 =x +y cos L . . . . (14)

Z -y sin L/sin 0 . . . . (15)

Also let P= [cos 0(1 - z 2) -x]/s in 0 . . . . (16)

Then the required correction (X- L) is given by

(X - L) =fzO - 88 z{0 (1 - 5z z) - 2P (202 - sin z 0) "~ +(1-z 2) s in0cos0}

( (1 - z 2) [2cos20- 1 -zZ(2cos20 +7) ] ' ) +-l~f3z sin 0 cos 0 L "~ + 8Ps in+ 8 p2 sin 20 cos0]0 [ - 1 +Z 2 (3 +2 tan2 0)~.j . . . . (i 7)*

- ~6f3z0 { 1 + 1 4z 2 - 3 1 Z 4 + 8P ( 1 - z 2) sin 0 cos 0 - 1 6P 2 sin 2 0} - 89 (I - 9z 2) + 89 3 { _ z 2 (1 - z 2) - 3Pz 2 cot 0

+ 2P 2} +. . . J * The terms in f 3 would not normally be used. They are given here so that the maximum

effect of neglecting them can be found.

14

LONG GEODESICS ON THE ELL IPSOID

This is a pecul iar formula as 0 can have all values from 0 to ,~: also P--+oo as 0--+r=. So it appears that although (X- L) cannot exceed fr= (radians), formula (17) may be indeterminate as some of the terms .+co as 0--+~.

6. First, it may be noted that x and z are both less than unity, so P sin 0 will never cause any trouble. But there are also terms in PO z, p203 and P03 cot 0. Since 0/sin 0.+1 when 0-+0, none of these will cause trouble when 0 is small. But when 0 is large and --+180 ~ the formula becomes indeterminate. I t is impossible to state precisely when the formula breaks down, but it seems likely that provided 0< 170 ~ (say), and using only the terms in fandf z, formula (17) will give (X -L ) to within about 0".005 or better. I t is suggested that the best method is to use formulae (1), (13) to (17) to obtain the first approximat ion to (X -L ) and then use (1) to (12) to obtain the precise result.

7. The computat ion of the distance is direct once the auxi l iary spherical tr iangle has been solved with sufficient precision. Let s be the geodesic distance and let u 2 =e12 cos 2 0r then the distance is given by:

s/b =Boa +B 2 sin a cos 2a m +B 4 sin 2or cos 4am +B 6 sin 3cr cos 6~., +B s sin 4~ cos 8Ore + . . . . . . . (18)

in which the coefficients are given by

Bo=I + 88 - 6-3~u4 + 2 ~ 6u'6 _ T6"5"g'~ u '175 -8

B2 = - 88 -t--11-6u4 - 31-115u6 -L'-2048-35 -8

- - 35 "8 B 4 = - T22-gu 4 +y~2zt 6 8192 u

B6 = - T~ 88 6 + r~u8 B8 = - ~ u 8

I

. . . . (19)

The max imum effect on s of any term in uS is less than half a mil l imetre, so that all can be neglected. Hence use the following simplified coefficients:

B o = 1 + ~u 2 - 3,~u4 + 2556 u6

B2 = - 88 + qtlu4 - -5!~s ~ . . . . (20)

B4= - 118//4 +-K~2 u6 |

3 B6 = - a-~-eu6 The max imum effect on s of the term in B 6 is only 1-3 mm so if absolute accuracy to the mil l imetre is not required this may be neglected also.

8. I f any of the formulae (I) to (5) do not give a sufficiently exact deter- minat ion of the required element, alternative formulae must be used. (1) always determines U1 and U2 well, but if either Pt or 92 are greater than 45 ~ use

cot U 1 =~ cot 91, or cot U2 =b cot 92 . . . . (21)

to avoid extra figures which provide no extra accuracy. I f a is near 0 ~ or 180 ~ it will not be well determined by its cosine; similarly if 0q or 0c 2 are

15

NOTICES SC IENT IF IQUES

near 90 ~ they will not be well determined by the sine. the azimuths first from:

cot {(oc 2 -0~1) =cot 89 cos 89 2 - Ul)/sin 89 2 + U1) cot 89 +~1) =cot 89 sin 89 z - U1)/cos { (U 2 + UA)

and then find e from

sin e/sin X =cos U1/sin oc z =cos U2/sin ~1

In either case find

. . . : (22)

. . . . (23) I f e is near 90 ~ then U1 and /-72 are small and sin U1 sin U2/cos2 ~x appears indeterminate. In this case instead of (5) use

cos 2e,, cos 2 ~ = cos e cos 2 0c - 2 sin U 1 sin Uz . . . . (24)

cos 2 ~ is found sufficiently exactly from (3). 9. The part icular solution of the direct problem which is now given was

first publ ished by McCAw2. I t does not require successive approximation. McCaw's formulae have been recast so that arcs are measured from the equator instead of the point of highest latitude, and extra terms have been included for greater accuracy.

P

7' \ / ~,'/(/~>"

Figure 2

Figure 2 shows a second auxi l iary spherical tr iangle which is used in the solution. In this triangle the longitude angle (BPC) is precisely the same as the angle BPC in Figure 1. The relationship between the az imuth angles of the great circle ABCD at A, B and C is given by

k cos (~) = cos

k cos (~1) =cos ~1

k cos (~2) =cos 0~ 2

For definition of k see formula (29). 10. Being given ~I, el, s the solution proceeds as follows:

b tan U 1 - - tan q~l - -a

sin ~ =sin el cos U 1

/ L (25) f . . . .

. . . . (26)

. . . . (27)

U2 = el2 COS 2 . . . . (28)

16

LONG GEODESICS ON THE ELL IPSOID

k2=(1 +u2) / ( l +e12 ) . . . . (29)

tan G 1 =k tan ?l/cos ~1 . . . . (30)

K"=~/ ( l+u2) /bs in 1" . . . . (31)

y =KC o s . . . . (32)

y I=G1 -C2 sin 2G 1 +C4 sin 4G 1 -C 6 sin 6G 1 . . . . (33)

2'2 =2"1 +'f" 2Tin =Y1 +Y2 . . . . (34)

G =y + O 2 sin 2" cos 2Tm + D4 sin 2u cos 4y~ +D 6 sin 3y cos 6y . . . . . (35)

G 2 =G 1 +G. 2Gm=G 1 +G 2 . . . . (36)

It may be noted here that formula (35) could be derived from formula (33) to give the arc G, but the arguments would be G and G~, still unknown. This difficulty could be overcome by solving by successive approximation, but this is a tedious process. McCaw solved the problem by using Lagrange's Theorem to change the unknown arguments G and Gm to the known ones y and y,,. For McCaw's solution of the direct problem, therefore, successive approximation has been entirely eliminated (see ref. 2, p. 348).

The coefficients of (32), (33) and (35) are given by:

C0=l 3U2 + 3__94U 4 -- ~T6 u133" 6--1- "i~-3--8--g u74"91" 8 "1 cz = ~u2 - 3 , ,4 _~ 11 , . 6 _1 , , . a [

. -1 -o - f ' s - - 2 -o -4~u !

C4 15 4 ~5 6~4os s ~ (37) = -z -g -6u - - - zw-6u 7- -8--f-9--s . . . . C6 = 3 5 _ . 6 3 o 7 2 ~ _ _~9_~j ,u 8 I Ca =1 ~_~,8 D2 = 3u2 -8"-3"4 ~ 1024" _213_. 6 _ 255f fu 8 D4 = _f21gu4 Zl__,, 6 _t. _!15 9_%, 8 L

/

-128~ - l~ss~ c . . . . (38)

The terms in C a and D a may always be omitted as inappreciable. The terms in'uS in C2, C4, C6 and D2, D4, D 6 are appreciable if accuracy to the millimetre is required (maximum effect %0".0001), otherwise they may be negIected. The maximum effect of the term in u s in Co is 0".0006 for a line half round the world. Hence, if a precise check to the millimetre is required between direct and reverse formulae, all terms in u a in (37) and (38) except for C a and D s must be included.

11. To complete the solution, we have

sin ?z = sin G 2 cos ~./k . . . . (39)

COS ~2 = k cot Gz/cot ?2 . . . . (40)

cos X = [cos G -s in p~ sin ?/]/cos ?1 cos 72 . . . . (41)

2 17

NOTICES SCIENTIFIQUES

(X -L ) =fs in ~ {Eoa -E2 sin G cos 2Gin +E4 sin 2G cos 4G~ - E 6 sin 3G cos 6Gin} . . . . (42)

E 0 = 1 - 88 (1 +f+f2) cos 2 ~ +1%f2 (1 +g f ) cos4 e -

~2~f~ cos ~ E 2 = 4if (3 + 5 f+ 7f 2) cos 2 ~ _ f2 (I + ~f ) cos4 0~ + 23 56 6sf 3

cos6= . . . . (43)

E4 = A f2 (1 + y f ) cos4 ~ - cos 6 E 6 =7@sf 3 COS 6 ~z

Note that E 0 is precisely the same as A 0 from (11). As before it is found that all terms in f 3 in the E's can be omitted as the max imum effect is less than 0".00001 and the following simplified forms may be used:

g o =A o = I -4 i f (1 +f ) cos 2 ~ +-~f2 COS 4 ~. ~]

E 2 =~f(3 +5f ) cos 2 = _ f2 cos4 = Ct . . . . (44)

E 4 =-~2f 2 cos 4 oc J

12. In the cases when q02 is near 90 ~ or when ~2 is small or near 180 ~ instead of (39) to (41), use Napier 's Analogies for the triangle CBP to find the angles at B and P, i.e. 180 ~ - (0@ and X and then find ?z from

cos q~a =sin (51) sin G/sin X . . . . (45)

I fX is near 0 ~ or 180 ~ use instead of (41)

sin ?, =sin G~v/ik2 - cos 2 ~l) /k cos q0 z . . . . (46)

13. Some confusion has often been caused in the past because parameters of the spheroid (Figure of the Earth) have been given which were not mutual ly consistent. Two parameters are sufficient to specify the shape and size of a spheroid completely. I f a and b are the major and minor axes respectively, modern practice is to define the spheroid by specifying a and r (the reciprocal of the flattening). Other parameters are then defined by:

1 f= (~ - b ) /~ =-

r

~2 = (42 _ b2) /42 = 2 _ _ 1"

i r 2

2 3 4 e12 = (a2 - b2)/b 2 =r +7 + r3 + . . . . . . . . (47)

1(1) n=(a-b) / (a+b)=~+ ~; + ~ + . . . .

1 b/a = 1 - -

r J

These are the simplest formulae to use for computat ion of the parameters, in terms of ( l /r) and powers thereof, with normal desk calculating machines.

14. As a pract ical check on the formulae given here, the following

18


examples have been computed and checked by use of both the indirect and direct formulae. ANDOYER'S tables of the natural sines, cosines, tangents and cotangents (15-figure) were used, taking the tabular quantities to 11 figures and all angles to the 6th decimal of a second: results were finally rounded off to the 5th decimal of a second and the nearest millimetre. Three values of (X- L) are given in each case: the first, rigorous from formulae (10) and (12); the second, by SODANO'S method from formula (17), neglects terms in f3 ; the third, also from (17), includes the terms in f3. Unless otherwise stated, results from the direct and reverse formulae agree to the 5th decimal of a second.

(a) McCaw's example (ref. 2, p. 158). Bessel Spheroid;

a =6377397"155 metres, r =299.1528128.

q~x +55~ 45' N; 7z -33~ 26' S; L+ 108 ~ 13' E

Rig. (X-L) + 14' 16"-62592 ~ 34 ~ 04' 48"-38630

Sodano 2 + 14' 16"-63006 ~1 96~ 36' 08".79960

Sodano 3 + 14' 16".62596 ~z 137~ 52' 22".01454

s 14110526-170 metres.

The errors of Sodano 2 and 3 are 414 and 4 in the 5th decimal of a second. The remaining examples are all on the International Spheroid;

a = 6378388 metres, r =297-0.

(b) q~l +37~ 19' 54".95367 N;

L +41 ~ 28' 35".50729 E

Rig. (X- L) + 05' 53".23775

Sodano 2 + 05' 53"-23741

Sodano 3 + 05' 53".23773

q~z +26~ 07' 42".83946 N;

52 ~ 25' 10"-99010

~l 95 ~ 27' 59".63089

~2 118~ 05' 58"-96161

s 4085966-703 metres.

The errors of Sodano 2 and 3 are 34 and 2 in 5th decimal.

(c) ~1 +35~ 16' 11".24862 N;

L + 137 ~ 47' 28".31435 E

Rig. (X-L) +03' 15".08310

Sodano 2 +03' 15"-08275

Sodano 3 +03' 15".08310

q~2 +67~ 22' 14"-77638 N;

12 ~ 48' 37".53647

el 15~ 44' 23"-74850

~2 144~ 55' 39"'92147

s 8084823"839 metres.

Residual longitude error is 3 in 5th decimal. The errors of Sodano 2 and 3 are 35 and 0 in 5th decimal.

(d) q~l + 1 o 00' 00"'0 N; q~2 -0~ 59' 53".83076 S; L + 179 ~ 17' 48".02997 E e 88 ~ 35' 17"'52180

19

NOTICES SCIENTIFI QUES

Rig, (X- L) + 36' 19"'96992 el 89 ~ 00' 00"'0

Sodano 2 +36' 20".26224 ~2 91~ 00' 06".11733

Sodano 3 +36' 20".27254 s 19960000.000 metres.

The errors of Sodano 2 and 3 are 0".29232 and 0".30262.

(e) q~l + 1~ 00' 00"-0 N;

L ~- 179 ~ 46' 17".84244 E;

Rig. (X -L ) +03' 07"-83471

Sodano 2 +02' 51".90751

Sodano 3 + 03' 11".95949

q~2 + 1~ 01' 15".18952 N;

4 ~ 59' 57".26995

el 4~ 59' 59"-99995

0~ z 174 ~ 59' 59".88481

s 19780006-558 metres.

Residual errors of latitude, longitude, az imuth are 2, 0, 1 in 5th decimal. The errors of Sodano 2 and 3 are 15".92720 and 4"-12478.

15. I t should be noted that there is one pecul iar ity of geodesics which approach 180 ~ of arc or halfway round the world. Take the case of two points on the equator, 180 ~ apart in longitude. I f these two points were on a sphere, it is obvious that there is an infinite number of great circles through both points, and all of the same length. On the spheroid, however, there are only two geodesic [sic] arcs, corresponding to great circles, which pass through two points on the equator, exactly 180 ~ apart. But the mer id ian arc is shorter than the equator ial arc through the two points, and it is, therefore, the only true geodesic on the shortest line definition of a geodesic. Consequently, if two points are situated near the equator and are separated by nearly 180 ~ of longitude there is a certain ambiguity as to what is meant by the geodesic between them. There is a full discussion of this po in t in an article 'The distance between two widely separated points on the surface of the earth'.3 This article is a review of another article of the same title by W. D. LAMBERT. 4

16. The two examples (d) and (e) show that Sodano's method breaks down when the arc between them is nearly 180 ~ Sections 5 and 6 indicate that the main cause of the trouble is the function P which -+Go as the arc --+180 ~ An attempt was made to get over this difficulty by splitt ing the line into two parts at the point of highest lat itude (where the az imuth is 90~ The function P for each part of the line is then zero. This does not seem to help, however, as this point is unknown: in fact, once this point is known, the whole solution of the inverse problem follows very simply. Sodano's method may also be used to obtain formulae, similar to (17), for (e - 0) and for sin e. Both, however, contain the function P and become indeterminate for arcs near 180 ~ It would appear then that all we can do about this is to define more precisely the useful limits of Sodano's method, by computing a sufficient number of examples, so that the max imum error under certain specified conditions can be stated.

17. In the past, the problem of long lines on the earth has frequently been approached by the use of plane sections, since the geometry was

20


(supposedly) easier to understand than that of geodesic curves in three dimen- sions. The difference in length between a geodesic and a plane curve is very small, but the difference in azimuth can be quite appreciable for lines over about 500 miles. If, for example, it was known that radar waves followed a plane section round the world, there might be some justification for plane section computation: in the absence of much real evidence on this or similar lines the geodesic approach is theoretically more correct and actu- ally simpler in practice. It seems probable that the method advocated by Levallois and Dupuy for computation of long lines is the best, provided that tables are made for various Figures of the Earth in sexagesimal units.

18. For lines less than 500 miles (800 km), in latitudes less than 75 ~ formulae are available using either plane curves or geodesicsS. The first two methods are based on plane curves; they will give precise results but require the use of 9 or 10-figure trigonometric tables. The other methods advocated are based on true geodesics and precise results can be obtained with 8-figure tables. For the inverse problem, the Mid-Latitude Formulae are undoubtedly the best as they do not require any successive approximation: the extension to fourth order corrective terms has been given here. For the direct problem the extension of CLARKE'S approximate formulae may be used: corrective terms have been given up to 6th order (spherical) and 5th order (elliptic), but many of these are inappreciable except in high latitudes. Formulae have also been given for the Puissant Series method up to 7th order (spherical) and 6th order (elliptic) terms. It is not recom- mended that these should be used for long lines as they do not converge sufficiently rapidly, but they are useful to obtain the possible errors from neglecting terms of any particular order.

19. The best geodetic tables available are those for Latitude Functions on the various spheroids (Natural values of the meridional arc; A, B, C, D, E and F factors; radii of curvature, R and N) produced by the Army Map Service in 1944. They tabulate the required functions to an accuracy of a millimetre at an interval of 1 minute of arc. These tables are invaluable for any form of geodetic line computations: they are clear, well set out and very easy to use as differences are given for 1 second. It seems strange that the A.M.S. computed these tables for every spheroid in common use, except their own (the Clarke 1866). If they could now see their way to producing the 1866 tables and, also, tables for the factors of the Mid- Latitude Formulae, while Levallois and Dupuy produced their tables in the sexagesimal system, the geodetic world would have available all the tables necessary for dealing with problems of long lines on the earth.

REFERENCES

1 SODANO, E. M., 'Inverse computation for long lines; a non-iterative method based on the true geodesic', Technical Report .No. 7, Aug. (1950).

2 McCAw, G. T., Empire Survey Review Vol. II, 156-63, 346-52, 505-8. 3 Empire Survey Review, Vol. VIII (1943), 172-6. 4 LAMBERT, W. D., 'The distance between two widely separated points on the

surface of the earth', J. Wash. Acad. Sci. 32 (1942), 125-30. 5 P~AINSFORD, H. F., 'Long lines on the earth: various formulae', Empire Survey

Review, Vol. X (1949), 19-29, 74-82,

21


SUMMARY

This article examines the practical application of formulae for computing long lines on the ellipsoid. The main aim is to eliminate the successive approximation generally required. For the inverse problem, this is achieved by the method of E. M. SODnNO, Army Map Service, U.S.A. An adapta- tion of a method produced by G. T. McCAw is used for the direct problem.

Results are given of five practical examples, including two which extend halfway round the world. Construction of further special tables is recom- mended to simplify the computations required by a problem which has an ever increasing application.

Rt~SUMI~

L'auteur, sp6cialiste bien connu de la question, examine et discute les diff6rentes m6thodes utilis~es ou pr6conis~es par diff6rents G6odfisiens, pour le calcul des lignes g6od6siques de grande longueur/t la surface de la terre.

ZUSAMMENFASSUNG

Der Verfasser--ein bekannter Fachmann auf diesem Gebiet--untersucht kritisch die von verschiedenen Geod~iten angewandten oder vorgeschlagenen Verfahren zur Berechnung sehr langer geod~itischer Linien.

R IASSUNTO

L'Autore, specialista ben noto dell'argomento, esamina e discute i diversi metodi utilizzati o proposti dai differenti Geodeti per il calcolo delle geo- detiche lunghe sulla superficie terrestre.

RESUMEN

E1 autor, especialista bien conocido en la cuestidn, examina y discute los diferentes m6todos utilizados o preconizados por diferentes geodestas para el cMculo de lineas geod6sicas de gran longitud en la superficie terrestre.

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long geodesics on the ellipsoid - hf rainsford

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