SIMPLIFICATION IN OBSERVATION AND COMPUTATION OF A TWO-STAR FIX

Transcription

1 SIMPLIFICATION IN OBSERVATION AND COMPUTATION OF A TWO-STAR FIX W IT H O U T USE O F TH E A L T IT U D E D IFFE RE NCE M E TH O D by Dr. Stjepo K o t l a k ic Assistant Director, Hydrographic Institute of the Yugoslav Navy INTRODUCTION This paper explains the changes introduced in the procedure for observation and coputation of a two-star fix using K o t l a r i c s Tables K 15 [1 ], published by the Yugoslav Hydrographic In stitu te1* 1. A practical exaple is given. These changes eliinate the need to take the copass aziuth of the first star, and to correct the first star s altitude for the tie elapsed between non-siultaneous observations of two stars. Instead, a shorter, sipler and ore accurate procedure for correcting a fix fo r elapsed tie is introduced. This is done directly by applying corrections fro the Main Tables to the tabulated Latitude and L H A Aries; fro the latter the Longitude is deterined by adding the tabulated SHA o f the second star and subtracting its GHA. Details of the Direct Method for coputation of the coordinates of a ship s position fro observations of two selected stars without the need to use the altitude difference ethod (Marcq St. H ilaire ethod) were given by the author in the July 1971 issue of the International Hydrographic Review [2 ]. It was there shown that the coputation of altitude and aziuths in order to obtain intercepts for plotting position lines, the intersection o f which gives the observer s position is not necessary if Tables K n are used, since the fix (the observer s Latitude and Longitude) is obtained directly fro the observed altitudes o f two selected stars. Also given in that article were the forulae used in the construction of Tables K u, together with excerpts fro the Tables and instructions for their use. A practical exaple was included. The way in which corrections have (*) Volue V N (Latitude 4(1 to 4!),'t0' North), published Volue IV N (Latitude 30 to 39.'l()' North), published Volue III N (Latitude 20 to 29 :i0' North), published Volues II N & I N, covering 20 North to the Equator, are in press.

2 to be applied for non-siultaneous observations in order to obtain a fix was also deonstrated, the principle being the sae as if siultaneous observations were taken. As regards non-siultaneous observations, there is still a need to correct for the ship s run between the first and second star sights; Tables III A and B re ain in force, no difficulties having been discovered or criticiss received fro ariners. Even if the heading angle of the first star (i.e. the relative azi uth) is out by 5 the resulting error can not exceed 0'3. In the case o f the tie elapsed between the first and the second sight (i.e. the increase o f the Greenwich Hour Angle of the First Point of Aries) it was deonstrated that a correction has to be applied to the observed altitude o f the first star using the well-known for ula : diff. A lt diff- L H A x cos Lat X sin A Z. This gives the first star s altitude as if it were taken siultaneously w ith the altitude o f the second star. In order to copute this correction it was necessary to take the copass aziuth of the first star and to transfor it into true aziuth; the copass aziuth had to be taken i ediately after the second star sight. F or si plification, i.e. to do away w ith interpolations, sign rules, etc., the author constructed a special auxiliary table (Tables IV A and B) which rendered this coputation considerably shorter and sipler than if the for ula given above were solved step by step. However, in spite o f this si plified table for correction of non-siultaneous observations, one difficulty has reained. This is the taking o f a correct copass aziuth of the first star. The new procedure described in the present paper eliinates this drawback, and thus both the coputation of the fix and the observations o f the two stars are shortened and si plified. EXPLANATION OF THE NEW PROCEDURE In the Direct Method, the need to take the copass aziuth of the first star was an extra step in the non-siultaneous observations o f two stars, one not required in the St. H ilaire Method. M oreover navigators have coplained of the difficulties in taking the bearing of a star at high altitudes. The accuracy of such an aziuth is thus liable to error. Considering that the non-siultaneous observation o f stars is the ost coon procedure in celestial navigation, one ust realise that in the procedure of correcting the altitude of the first star with the above forula (using Tables IV A and B) an error of 1" in the observed aziuth can produce an error in the corrected altitude o f 1', leading to a possible position error o f greater than 1 nautical ile. This is because the error in altitude of the first star (d H o,) is reflected both in angle B and in the parallactic angle ni; and the error in the parallatic angle (d ) is further reflected both in the coputed Latitude (L a t) and in the Local Hour Angle of the second star (LH A.,), i.e. in the Longitude. The follow in g for ulae explain this :

3 db = dhoj. sec H o2. cosec (A Z l A Z 2) ni = A + B, ni = A B, ni = B A, ni = 360 (A + B) dlat = dni. cos Ho2. sin A Z 2 d L H A 2 = dni. cos H o2. cos A Z 2. sec Lat These for ulae were derived in the present author s earlier work New Methods of Ship Position Finding fro Celestial Observations (1955) [3 ]. As was noted on pages (in Serbo-Croat language) of the second edition (1973), an error of + 1' in H ox leads to an error in Latitude o f If 1, and in L H A 2 to an error o f + 0'2 in Longitude. It is i portant to note that with the new procedure here described for direct coputation there is no possibility of such aziuth errors, since the need to take the first star s copass aziuth and to ake the corresponding correction to its altitude based on this aziuth have both been eliinated. Thus the auxiliary Tables IV A and B are no longer necessary. Instead, the new procedure introduces corrections for the coordinates o f the observer s zenith (the Latitude and L H A Aries) taken directly fro the Main Tables in K n. This correction is based on a single arguent, i.e. the elapsed tie between sights o f the first and the second star. It is obtained fro Table IV C as a correction for the change in difference of sidereal hour angles (diff. SH A) of the two stars w ith its sign, which is ultiplied (through Table V - Multiplication Table) by the correction indices (given in the Main Tables) to allow for a change o f sidereal hour angles (ISU ). Table IV C and an exaple of this ethod of coputing a fix are shown below in detail. Firstly, however, it ust be shown w hy the correction of the first star s altitude for the elapsed ti e is identical with the correction to the Latitude and LH A Aries, both representing the sidereal hour angle difference fo r the tie elapsed between the two star sights. As is w ell known, the difference between the sidereal hour angles of two stars is nu erically equal to the difference between their Greenwich hour angles (SHAj^ SH A2 = GHAj GHA2). Accordingly, if 24s has elapsed between observations of the first and the second star then an increent of 31'1 of GHA Aries is obtained fro the Nautical Alanac. This indicates that the Greenwich hour angle o f the first star has been si ilarly increased; consequently the difference between Greenwich hour angles for these two stars aounts to the sae value. The result would be the sae if the difference between sidereal hour angles o f these stars were changed by the sae aount of 31'1. Thus corrections to the coordinates of the observer s zenith, i.e. corrections to the Latitude and L H A Aries (Longitude), can be coputed to allow for the change in LH A of the first star in the sae anner as for the change in its sidereal hour angle. It w ill however be necessary to pay attention to the signs of the Lat. and L H A Aries corrections since in Tables K n the indices of the corrections for changes in difference of sidereal hour angles (the ISU indices) are tabulated for + 1' increents of tabulated difference o f sidereal hour angles (SH A! SH A2), always provided that declinations and altitudes of both stars reain unchanged. Further ore, the elapsed ti e between sights of the first and the second star transfor ed with the help of the

4 Nautical Alanac into an increent of GHA Aries this is seen in the sky as apparent otion of the star fro East to W est can be expressed as an increase or decrease in the eridian angle of the first star, depending on its aziuth. This eans an increase or decrease in the first star s altitude, and consequently the conditions eployed when tabulating the sign fo r ISU indices are not the sae. Hence, Latitude and L H A Aries corrections for increase of Greenwich hour angle of the first star w ill not always have the sae sign as that of the increase in the difference of tabulated sidereal hour angles SHA, SH A 2. If ISU indices for Lat and L H A corr., with their signs, are e ployed then the sign for the correction to the difference of sidereal hour angles ust also be known, so it is now7 given above the tabulations in Table IV C. W ith the elapsed tie as the entering arguent, the nuerical value is found. Fro a study of the celestial sphere a very siple rule for this sign has been drawn up. This is : the sign for corr. diff. SHA is always +, excepi when the first sta r is west of the second one, when it w ill be. Copass aziuths of stars are not necessary for this purpose because in the Main Tables K lt the aziuths are tabulated below the star naes. Thus at a glance it can be seen if A Z j is greater than AZ.,; it is evident also fro the star observations whether the first star is west or eastward of the second one. Table IV C therefore yields corr. diff. SHA with its sign, and this has to be algebraically added to the value diff. SHA which has been obtained by subtracting the (SHA] SH AL.) value shown at the top o f the pages in Main Tables K,, fro the (SH A, SH A 2) value read fro the Nautical Alanac, in order to deterine a final diff. SHA. By application of the ISU indices fro the Main Tables for Lat. corr. and L H A corr., this final diff. S IIA yields corrections to both Latitude and L H A Aries, with their signs; the ultiplication Table V is used for this operation. Th e above explanations ay see coplicated to a ariner, but they are intended pri arily for those interested in aking a deeper study o f navigational ethods, In fact, the advantage of the i proved ethod is lhat the ariner has no need to go into the theory because the Instructions for Use together with the solution of an exaple printed at the end of the present paper enable an easy understanding of this coputational ethod for a fix. The use of Tables K n is further facilitated by a cardboard insert of the Instructions fo r Use, including a saple solution printed in five colours. Th e exaple given on page 105 o f y 1971 paper [2 ] shows that for the tie elapsed between observations of Antares and Vega the correction of the first star s altitude was 21'1 (see Table IV A and B), with as entering arguents the true aziuth o f Antares (136 ) and the dead reckoning latitude (12.2). This correction (diff. Ho, = + 21'1 ) when ultiplied by the tabulated indices IV, (Indices of corrections to the Latitude and Local Hour Angle of Aries for a change o f + V in the first star s altitude) yields the follow ing corrections to the Latitude and L H A Aries ; diff. Index LAT. LAT. corr. diff. Index LHA7 LHA corr. Ho, (V j ) + 2 1'. 1 x - 80 = - 16'.9 Ho, (V.) + 21'.1 x + 62 = + 13'.1

5 Meanwhile, by the new procedure and the elapsed ti e o f 2 4 s taken fro the new Table IV C (see page 164) we obtain the correction to the difference of sidereal hour angles as corr. diff. SHA = 31'1. If this correction is ultipled by tabulated ISU correction indices the follow ing corrections to the Latitude and L H A Aries are obtained : diff. IND. LAT. diff. IND. LHA? LAT. corr. LHA corr. SHA (SU) - 31'.1 x + 54 = - 16'.8 SHA (Sin - 31'. 1 x - 42 = + 13'.0 This exaple worked by the two different ethods, (a) the auxiliary Tables IV A and B, and (b) the new auxiliary Table IV C, shows that the corrections to the fix for elapsed tie are very nearly identical in both cases. W ith Table IV C there is a difference of O'.l for each correction, copared with the Table IV A and B results, but the for er represents ore accurate values since Table IV G applies elapsed tie corrections directly to the coordinates of the observer s zenith found in the Main Tables K u. This is better than the indirect procedure using Tables IV A and B to find an approxiate solution for diff. Hoj = diff. L H A X cos Lat. X sin AZ, where the aziuth is liable to error. The new procedure using Table IV C is further ore sipler both for observations and for coputation. Stationary observer It is now considered useful to explain w hy there is a need to correct the Latitude and L H A Aries (i.e. coputed observer s position) if the ship does not ove during the elapsed tie between the observation of the two stars. It ust be reebered that Tables K lt are entered with the two star altitudes not taken siultaneously, and thus the fix obtained fro Tables K u is erroneous since the change in difference of their sidereal hour angles for the ti e elapsed between the observations has been neglected. Thus the corrections to Lat. and L H A Ariës obtained by use of Table IV C and ISU Indices fro the Main Tables in fact represent corrections for this neglected error rather than for oveent of the actual position of the observer. This new procedure for correcting a two-star fix for the tie elapsed between the observations has necessitated changes in the current Instructions for Use for Tables Ku, and alterations have accordingly been ade in paragraphs 3, 4, 5, 6, 9, 10, 11, 14 and 16. The new shortened Instructions for Use are given in Serbo-Croat and in English (reprinted overleaf) accopanied by a W ork For printed in five corresponding colours, and show the solution of the follow in g exaple o f the two-star fix by the Direct Method using Tables K n. Exaple On June 12, 1988, the zone tie 19h50 dead reckoning position of a ship is Lat 'N, Long ' E. The ship is under w ay, the speed

6 SHO RTEN ED INSTRUCTIONS FOR USE Preliinary work and work with Nautical Alanac 1. U sin g the Approx. GMT for the intended observation and D R Long, copute the Approx. LH AT. 2. With the approx. DR Latitude a n d L H A T found in Step 1 turn to the pages in T a b le s K l l. 3.C h oose one o f the four pairs o f stars given either on the le ft or the right page and take a sight on the first and then on tne second star. 4. O bservations o f the tw o stars can be ade with any tie1interval up to 4 inutes. T h e second sigh t can be taken within a short interval if the altitude o f the second star is easured approxi ately in advance. If the 4 - in u te interval is exceed ed, and the observation s cannot be repeated, the first star's altitude should first be corrected fro T a b le III for 4 inutes and then for the reainder o f the ti e (See Step 11). In that case T a b le IV C in Step 14 w ill be used in the sae w ay. 5. For non si ultaneous observation s take, for the first star, aziuth re la tive to course (heading a n gle, 5 accuracy s a tis fie s ). In addition take note of the speed o f the ship. 6. S olve the proble by inserting the values in their re s p e c tiv e places on the work for. (T h e work for book let can be purchased). 7. T h e p osition is being solved for the tie o f the second s ig h t. U sing the GMT for the second sigh t find the G H A T and the coordinates for both stars (D l, D2, S H A l and SHA2). 8. Find the valu e S H A 1 -S H A 2 by subtracting the s aller valu e. If the result exceed s 180 subtract it fro Find the GHA o f the second star (G H A2). 10. Deter ine the d ifferen ce o f GMT between the tw o sigh ts (d. G M T).!!. C orrect the sextant altitude o f the first star ( h s l) by T a b le I (or by T a b le II if a bubble sextant was used) to obtain the observed a ltitude H o i. If th e tw o sights w ere taken at d ifferent ti es (s e e Step 4 ),the valu e H o i should now be corrected by T a b le III to obtain H o i cortd. T h e sign s are given in the T a b le and no interpolation s are needed. i I. C orrect the sextant altitude o f the second star (h s2 ) b y T a b le I (or b y T a b le I I)t o obtain the ob served altitude Ho2. Use of the Main Tables and Multiplication Table 13.R eenter the Main T a b le s on the sae page and stars, but in coluns H o i and Ho2, using the line showing figures nearest to the observed altitudes H o i (or H o i cortd.) found in Step 11, and the Ho2 found in Step 12. Fro this line take out: (a ) the tabulated values H o i and Ho2; (b ) L H A A r ie s ; (c )th e correction in d ices I V 1, IV2, ID 1, ID2 and ISU, with their sign s, fro the coluns In d exla T. C orr. and Index L H A C o r r. (a ll the values printed without a sign a r e +), Fro the top of the Table take: L a t., D 1, D 2 and S H A 1 S H A Find the d ifferen ces, together with th eirsign s: H o i (cortd.) - H o i tabulated = d iff. H o i; H02 - H02 tabulated = d iff. H0 2 ; D i - D i tabulated = d iff. D i; D 2 - D 2 tabulated = d iff. D2; (S H A i - SH A2) - (S H A i S H A 2) tabulated = = d iff. SHA, then add corr. fro T a b le IV C. 15. Insert the signs o f the L A T. and LH A corrections in the work for after deter ining the by eans o f the signs for the differen ces obtained in Step 14 and of the sign s of the corresponding correction indices taken fro the Main T a b le, in Step 13 (c ). When these differen ces are p o sitive the signs for the corrections are the sae as those for the correction indices in the Main T a b le. For the negativ e d ifferen ces the signs are opposite. 16. Enter T a b le V with the correction index IV l on the le ft and d iff. H o i at the top to find the required correction. Th e correction to the tabulated Latitude is obtained by using the I V l o f the Index L A T. Corr. Then using the I V l o f the Index L H A Corr. the correction to LH A A ries is obtained. Do the sae for the IV2 and d iff.h o 2,ID l and d iff. D l, ID2 and d iff. D2, ISU and d iff. SHA. No ental interpolation is required. 17. Add the p o s itiv e corrections and subtract the su o f the n egative corrections, in order to obtain the true valu es for LATITUD E and LHAT? 18. T o the L H A T add the tabulated SHA2 fro the top o f the Main T a b le to obtain the LH A2. Deduct 360 if n ecessary. 19. Subtract the G H A 2 (Step 9) fro the L H A 2 to obtain the LO N G IT U D E with its sign ( - f o r W and + for E ). If the result is greater than 180 subtract it fro 360, changing the sign.

7 WORK FORM FOR DIRECT FINDING OFATW O -STAR FIX WITH KOTLARIC'S TABLES K11 Date 12 June 1968, zone tie 19^ 50, dead reckoning Lat N, Long. 17 1st star's AZ relative to course 16, speed 16 kn, zone description I*1, 10 E, h. eye 11, 1C O. Approx. LA T. 42 N 1. SPICA 2. REGULUS Greenwi ch date 12. VI Approx. GMT 19h 00" Approx. GHAT corr Approx. GHAT DR Long. -, E t Approx. L H A T GMT 2 - GMT l d. GMT 1st star's R elative AZ Main Tables 18h s l 41.0s 16 Tab. I, II, III and IV Tab. V + d if f x ± IND = + corr. d iff x i IND = +corr. CT 18h s CE GMT l 18h s hs C 0 Tab. 1 -e rit Ho l ' rtab. Ill Ho 1 cortd tabulated d iff. Ho Dec tabulated diff. D l SHAl SHA SHA1-S H A tabulated d iff. SHA Tab. IV C d iff. SHA ' CT 18h s CE GMT 2 18h 59^ 46.0s hs ' 1C 0 Tab. 1.0*-H- 7.2 Ho tabulated d iff. Ho2-6.0 Dec tabulated d iff. D ' G H A T ' corr SHA GHA d iff. IND. LAT. IND. LHA Aries LAT. + corr. corr. LHA + corr. _ corr ' N H oi (V l) ' _ 15.1' Ho2 (V 2) _ D l _ 0.2 D _ SHA (SU) _ 2 _ L A T N L H A T + tab. SHA2 LHA ' GHA LONG ' E

8 EXCERPT FROM TABLES Kn - MAIN TABLES 42 N SPICA* REGULUS* SU1 - SU2 * * 2 FI (LAT.) * *1 * (SMA1-SHA2) A A * (INDEX l a t * c o r r. ) < INDEX LHA C O R R * 1 ARIES D 2 * «8 INDEKSI COPRAVAKA FI INOEKSI POPRAVAKA MS MS (LHA) VI /HOI) V2 1H02J IVl I V2 101 JD2 ISJ IVl JV2 ID1 102 I SU 0 t 0 I ' ' ' '* * » ' ' * 9 * ' T A B L. I V C CORRECTION TO THE diff.sha FOR THE TIME ELAPSED UNTIL SECOND STAR SIGHT ( c o r r. d i f f. S H A a lw a y s e x c e p t " " w h e n th e f ir s t s ta r is w e s t o f t h e s e c o n d ) 0 (c o r r. ) 0 ( c o r r. ) 1 (c o r r. ) 1 ( c o r r. ) 2 (c o r r.l 2 ( c o r r. ) 3 ( c o r r. ) 3 t s / s ' s > s f $ ' s ' s ! (c o r r. ) N o te : W h eth er th e first star is w est o f th e secon d it is also sh ow n in M ain T a b les b y t a b u la t e d a z i u t h s, i. e. w h e t h e r A Z 1 is g r e a t e r th a n A Z 2.

9 16 knots, zone description l h. The true position is wanted by observations of two stars at approx. Z T 20h (GM T 19h), and solved by Tables K u. Fro the Nautical Alanac with approx. GMT and D.R. Longitude the approx. L H A y 203 is obtained. W ith this and approx. Lat. 42 N the Tables K u are opened and the pair of stars SPICA - REGULUS chosen for observation. The sights are taken, firstly on Spica hs! 37 19'1, chronoeter tie CTX 18h5545.5s, then on Regulus hs f7, CT2 18h Chronoeter error on GMT is e. Spica s heading angle (aziuth relative to course) is 16. Index correction o f the sextant is 0, height o f eye 11. Find a Fix at the tie of the second sight. Note: The necessary excerpts fro Main Tables Kn and Table IV C are given opposite. The sae exaple by the previous ethod, i.e. using the Table IV A and B, was shown in Volue V of Tables K n, page PRINTED AMENDMENTS The new procedure for coputation of a two-star fix fro nonsiultaneous observations, as here described, w ill be included not only in Volues I, II and III of Tables K n now in press, but also in Volues IV and V already issued. The latter already contain the special five-colour cardboard inserts in English and Serbo-Croat; owners o f these volues ay now obtain the revised inserts free of charge fro : The Hydrographic Institute, S PLIT, Yugoslavia, to enable the to use this new and shorter procedure for the observation and coputation o f a two-star fix using any volue o f Tables Ku. CONCLUSION The essential difference in the new procedure is the eliination of the need to take the copass aziuth o f the first star; this answers the ain criticis of Tables K n. Also, the new procedure, besides being shorter and sipler, is ore accurate. W hat now reains to be done is the process of fa iliarizing navigators with this new ethod and of breaking down the barrier of conservatis which is a fairly coon hazard for all innovations. However, it ust be reebered that Tables K n enable calculation of the fix coordinates direct fro the Tables theselves, and eliinate the precise graphical work on the plotting sheet or an approxiate sketch, and the atheatical process of finding the coordinates for the intersection o f the position lines required in the St. Hilaire ethod. This atheatical

10 process becoes ore coplicated when position lines are plotted fro different assued positions as is usual in navigational practice. In this respect the Direct Method is preferable to the indirect Altitude Difference Method. THE FUTURE W e live in an era of ever increasing applications for coputers, and the efforts being ade all over the world to introduce electronic calculators and coputers aboard ship cannot be ignored. Coputer reduction of celestial observations to lines of position w ill thus be possible, and the need for the different types of existing celestial navigation tables ay be diinished. In y estiation, the above described Direct Method of finding fix coordinates w ill be of ore interest than the St. Hilaire Method for coputer applications. Many different short ethod tables for altitude and aziuth coputation exist, and as far as I know create no difficulties for navigators. The fact that y own Tables K! (1958) [4] were in their third edition in 1971 testifies to their continued wide use (see reviews of these Tables [5 ], [6 ]). Accordingly, in y opinion, there is less need to replace the short ethod tables by ini-calculators or coputers to give altitudes and aziuths than to develop a coputer ethod giving directly the fix coordinates (Latitude and Longitude). W ith this goal in ind y book New Methods of Ship Position Finding fro Celestial Observations [3 ], 1973 edition, suggested the annual tabulation in the Nautical Alanac of about 50% of the data required for this Direct Method for coputation of a two-star fix ; the reaining calculations one angle (B) in the spherical triangle fro three known sides, and one angle (L H A ) and one side (Lat.) fro the other spherical triangle with two known sides and their included angle could be carried out easily with a celestial navigation coputer, or even a hand-held calculator. W ith Latitude and LH A coputed it w ill not be difficult to find the Longitude since this is obtained, with its sign, fro the Local and the Greenwich hour angles (Long. = L H A GHA). The application of this Direct Method to double observations of the Sun was also deonstrated in the sae book, and the solution of a practical exaple given. This would be yet another use to which one of these inexpensive pocket-size calculators could be put as part of the navigator s personal equipent. Research is at present being carried out into the applications of electronic calculators and coputers for work onboard ship in several different countries; in the U.S.A., Hewlett-Packard, and the Marine Division o f the Micro-Instruent Copany have both ade significant contributions to this new developent. However, only the future w ill show whether the new developents w ill eet navigators wishes to such an extent that they w ill finally be convinced of the need to change their conservative

11 ethods for sight reduction, packing away their reduction tables to be kept in reserve, and to take up the electronic box for their calculations. For the tie being, short ethod tables reain in predoinant use for solving a fix fro celestial observations even on board ships belonging to countries which lead the w ay in coputer production and' the applications of coputer techniques to the various kinds o f huan activity. Just how long the existing shipboard situation w ill continue is difficult to foresee; the choice had better be left to the user; in other words let us leave navigators the ajor responsibility o f deciding which o f the accessories available the tables or the electronic gadgets they w ill use for their onboard calculations. BIBLIOGRAPHY [1 ] K o t l a r ic, S. : Tables K n for two star-fix without use of altitude difference ethod, Vol. V N (Lat N ) and Vol. IV N (Lat N ). Hydrographic Institute of the Navy, Split, Yugoslavia, 1971 and [2] K o t l a r ié, S. : Two-star fix without use of altitude difference ethod. Int. Hydr. Review, Vol. X L V III, No. 2, July 1971, pp [3 ] K o t l a r ié, S. ; New ethods o f ship position finding fro celestial observations. Hydrographic Institute of the Navy, Split, Yugoslavia, 1955 and Also : Int. Hydr. Review, Vol. X X X III, No. 1, May 1956, pp [4 ] K o t l a r ié, S. : Tables K^. Short ethod of coputation of altitude and aziuth in astronoical navigation. Hydrographic Institute of the Navy, Split, Yugoslavia, 1958, 1963, [5 ] B o w d it c h, N. (1958 et post.) ; Aerican Practical Navigator, Chapter X X I - Coparison of various ethods o f sight reduction, articles 2112 and U.S.N. Hydrographic Office Pub. No. 9, W ashington. [6 ] Sa d l e r, D.H. & Sc o t t, W.H. : Book Review. New ethods of ship position finding fro celestial observations. S.M. K o t l a r ié, in : Journal of the Institute of Navigation, Vol. XI, No. 2, April 1958, pp London. [7 ] M o o d y, A.B. : Book Review. Tables K u. In : Navigation, J. of the Institute of Navigation, Vol. 18, No. 4, W in ter , pp Washington. [8 ] Sa d l e r, D.H. : Book Review. The double-altitude ethod, Tables K n. In : J. Navig., Vol. 27, No. 2, A pril 1974, pp London. (Manuscript subitted in English)