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ian reads: OcToBER II, 1900. THE MARINE RECORD. See LATITUDE AND LONGITUDE BY THE GREATEST ALTITUDE, AND BY EQUAL ALTITUDES OF A HEAVENLY BODY. . During the last two centuries vessels have been groping their way in the dark on the trackless.ocean on account of ‘defective, officially approved astronomical methods, and in “many instances it isa wonder that they escaped stranding despite gross errors in their reckonings. Depending for lati- tude principally on meridian altitudes, the speed of vessels ‘and the change of the declination of the observed objects “affecting altitudes, and consequently also latitudes, have never been taken into account, causing errors in position - which have brought many a vessel to grief. Latitude is an indispensable factor in the solution of every nautical-astronomical problem, With the latitude in error, the results of all reckonings for time, longitude, the varia- tion and deviation of the compass, etc., are also in error. The magnitude of these errors influencing position has never been ascertained, so that it justly may be said vessels have been groping their way in the dark. The change of position and the change of the declination during observations affect altitudes on and near the meridian sometimes considerably, for which no allowance is made by the methodsin use. Neither is the difficulty of noting the time for such altitudes taken into account. In observing altitudes near the meridian and noting the time by a watch, the nearer the meridian, the more indistinct becomes the time pertaining to a certain altitude, the change of altitude in one minute of time becoming so small as hardly to be noticeable in the instrument. Therefore, in observations for time, altitudes should be so far from the meridian that their change in one minute of time is easily!perceptible. Furthermore, as altitudes over the sea horizon are liable to an error of at least one. minute, the change in: altitude should beso large that an error of one minute in altitude has no great effect on the observed time. An error of one minute in altitude, when the change in altitude is only one minute in one minute of time, causes the latter to be one minute in error; a change of two minutes in altitude under the same conditions causes the time to be 30 seconds in error; a change of three minutes in altitude causes the time to be 20 seconds in error, andso on. Twenty seconds in time equal 5 minutes in longitude. Taking this to be the - greatest allowable difference from the correct longitude in, the following table shows the least hour angles at which ob- servations for time near the meridian should be made: TABLE I. ~ Showing the hour angles in time at which altitudes change - 3 minutes in-1 minute of time. Sy Gosduacle shail 4 frat wots Hanh F Dec'ination of « iffer- F Declination of the ‘ F ent name fr m Declina- same name as Supe latitude. Hon. latitude. 30° 20° 10° (ele. 10° 20° 30° M. S.|\M. SM. S.| M. S. |M. S.|M. S.J M.S. Io 34 40| 24 48] 16 12 8 4 o o|. 8 36] 18 24 20 43 24] 33 28] 24 48] 16 4o 8 36, 0 oO} 9 48 30 | 53 24] 43 24) 34 40} 26 32 | 18 24). 9 43) O Oo 40 65 52! 55 40] 46 52| 38 40 | 30 28] 21 48) 12 o 50 82 52| 72 32| 63 32| 55 30 | 46 52| 38 8] 28 12 60 |1I0 oj] 99 8] 89 44| 81 4 | 72 32) 63 32] 53°24 7O |167 56/153 56/143 8] 133 20 |123 48/113 52/102 52 The method to be considered farther on demands a re- striction of hour angles to about one hour, so that a line drawn in the preceding table from 4o degrees of latitude in column 2 to 60 degrees of latitude in column 7 shows about the limit of observations, and also shows that only a small margin is left in some cases to carry out observations for - time. Within about one hour on each side of the meridian, the -- differences of altitudes with the meridian altitude, and the hour angles, follow the simple law of the co-ordinates of a parabola. For, the equation of the reduction to the merid-_ 225 t? sin 11 ig ee ae 2 (tan bi tan c) in which H—h is expressed in arc minutes and tin time minutes; H denoting the meridian altitude, h the ex-merid- ian altitude, t its hour angle, b the latitude and c the decli- nation. 225 sin 1} I Putting ee =— 2 (tan b_ tan c) P P = 30,558 (tan b | tan C)Te Re es (1) and expressing H—h by y. jo Aa SSRN Aman Sen. hee Ben. ean(2) which is the co-ordination formula for the parabola, p being its parameter. Changing the beginning of co-ordinates from the vertex to a point of the curve a minutes East for which y = q, for- mula (2) transforms into t denoting the time from the beginning of co-ordinates. A change of position on the true course n at the rate of e miles per time minute, causes in t minutes a change in lati- tude and consequently in altitudes of t e cos n, and in the hour angle a change of tesinn. On account of the change 15 cos b in latitude alone equation (2) transforms into t tecosn y= —(2a—-t) P ce = —(2a—t He pe cos n) upper sign when latitude increases, lower sign when latitude decreases. A change in longitude alters the time so that __ t,esinn _e sin n ee be eh oat cn 15 cos b | +1. cos upper sign for westerly, lower sign for easterly courses. Putting fe) Sine I Se SEB EE eGR eo Se un aieean nao 4 te I5cos b ) and writing t for t, tg es y= jaete eo pe cos a} oe ea (5) " which is the general formula for the difference of altitudes near the meridian, with the altitude at the beginning of co- ordinates, on a change of position, from which a can be found, as y and t are known by observation. For any two altitudes whose differences with the altitude at the beginning of co-ordinates are y, and y, and the respec- tive time from the beginning of co-ordinates t, and t, we have the following expressions ti. y, = (2a— t, 4 pecosn) t, g ise and y, = —— (2a — t, g | pe cos n) from which is found P..You71 le ae 4 fe (yt. oe T pecosn] : g t,—t, For equal altitudes y, —‘y; = 0, therefore t, +t, 4 Pecos n a=g a velo eles e's (6) 2 2 By differentiation of formula (5) is found that y is a maxi- mum for Se 2a+ pecosn t— : 2g Denoting t for which y is a maximum by T, it follows that pecosn a=gTt 2 Comparing the last expression with (6) we find t, +t that..-— 2 F , proving the symmetry of the curve, and 2 the maximum ordinate to bein the center between t, and t,. As this is true for all equal altitudes, T equals half the | elapsed time between any two equal altitudes; the beginning of co-ordinates changing with the altitudes. This property of T renders it very easy of observation. Substituting the value of a formula (7) in formula (5) the latter transforms into ts y = P which is the general formula for equal altitudes; y is a max- imum when t = T or when p Subtracting T from formula (7) oP aT (Gos yor pe cosn or a— Tike) ae 2 esin n but asg —1 = 7, —— i ee 15 cos b esn € co: oe soy bees 15 cos b e 2 As the left hand side of the equation represents for ie + pe cosn —-—— is 2 the correction of T to find the position of the-changed meri- dian for the time indicated by T, and as T is-counted from pe cos n east to west, T+ — time T, the abscissa of the changed meridian, remains on the east side of the is : changed meridian; therefore, counting the correction from ee __ pe cos n> the meridian, its sign has to be reversed into es ———\§\-the PBs sik minus sign then indicating an easterly, the plus sign a wes- terly hour angle, as is generally the case. The reduction of the greatest altitude to the meridian i is t? obtained by the usual formula y —— in which t equals __ pe cos n 3 +. -, consequently the reduction 2 p.(¢)c0e nt a ice (11) 4 This correction is always to be added to the greatest allti- tude in order to obtain the meridian altitude, and from it in the usual way the latitude for the time or moment indicated by T, The object will be on the meridian for the greatest alti: pe cosn tude — minutes before or after the greatest altitude i is a2 observed. But asthe meridian also changes place during I pe cos n that time at the rate of —, ———— is the time required be- g 2g fore or after the obj ect is on the changed meridian, and the reduction in reference to the greatest altitude a {eet p (ecosn)? y = a Pp 2g 4 the same as the reduction of the greatest altitude to the meridian at the time T. The difference, however, is that this correction has to be subtracted from the greatest alti- tude in order to find the altitude at the object’s transit, the real meridian altitude. The hour angle of the greatest altitude in reference to the meridian atthe time T, = 2 PE COS Nar a rosa cas meran (12) s= Seis ees =e and ‘in reference to — meridian at the time of transit, PE COMM inary. setae eee te (13) sea 2g From which follows that the hour angle of the greatest alti- tude is a maximum when n is zero, that is, when the true course is north or south, and zero when n = go°, or the true course is east or west. ; Moreover, it is worthy of note that the hour angle of the greatest altitude represents half the difference of the hour angles of all equal altitudes near the meridian. JOHN MAURICE. Civil Engineer and Nautical Expert. CHICAGO, Sept., 1900. (To be continued in next issue.) eee ee EEE THE range finder is of great service as an aid to navigation. It is very well known that the distance of any object at sea, and more especially the distance of a light, cannot be esti- mated with any approach to accuracy. So much is this the case that in certain weathers experienced navigators fre- quently mistake a distant light-house for a near ship’s light and vice versa. The method sometimes adopted for deter- mining the distance of an object on shore by observing the change of bearing upon a known distance run, is, of course, not applicable at all when the light is within some points of the bow (the most important case) and in any case it re- quires a considerable amount of time and may involve the incurring of great danger. The range finder will determine the distance of a light in a few seconds, and the light can, when necessary, be kept constantly in view and the distance read as frequently as desired.