March 21, 1895] 



NATURE 



485 



LETTERS TO THE EDITOR. 



[The Editor does not hold himself responsible for opinions ex- 

 pressed by his correspondents. Neither can he undertake 

 to return, or to correspond with the writers oj, rejected 

 manuscripts intended for this or any other part of Nature. 

 No notice is taken of anonymous communications. ] 



Corrections of Maximum and Ex-Meridian Altitudes. 



In navigation the error inlroiuced by taking the maximum 

 altitude of a heavenly body for its meridian altitude is njt 

 sufficiently great to need correction when it is du; to variation 

 of declination alone, as it is then much within the probible 

 errors of observation. When, however, a ship is steaming at a 

 high speed, the error is considerably inireased by the variation 

 of latitude, especially when this is of opposite sign to the varia- 

 tion of declination. 



The formula giving the correction in seconds of arc to be 

 applied to the zenith distance of a body fo; red iction to the 

 meridian is 



x = Ch- (I) 



cos /cos 5 sin- 15' 



where, with the usual notation C : 



and 



2 sin (/^S) sin l" 

 h is the hour angle in minutes of time. Thu;, if : be the zenith 

 distance, we have the equation 



2 - C/;= = / q: 5, 

 the upper or lower sign being taken according as / and 5 are of 

 the same or of opposite sign. Since we may consider z, h, I, 

 and 5 as functions of the mean time t and C CJnitant, v.e have 

 on differentiation, if ; be a minimum, 



dt dt ^ di 



Let H and H' be the hour angles of the bjiy an I the ship's 



zenith measured from som; fixed meridian, anl let k, v denjte 



the northerly and westerly components of th; ship's velo;ity in 



dh dH dH' ^ dl Ai -f.v, K J ' 



— ._j Alsoif the bady s 



2Ch'^ = 



knots ; then 



— r- and , 



dt dt dt dt 



projection on the Earth's equator moves round at the rate ot 

 U knots, 



——/ ■— = u sec //U ; 



dt / dt 



and the above equation becomes 



2C/i— (I - »sec//U) = z; q: 5 



(2) 



Now this equation will assum; different forms according as the 

 body under consideration is the Sun, a star, or the Moon. For, 

 since the motion of the ship is expressed in mean solar time, the 

 motion of the body must likewise be expressed in that time. 



The unit o( time being a minute of mean time, and the unit 

 of arc a second of arc, we have for the Sun 



~= I and U = 900, 

 dt 

 so that (2) becomes 



- 2Ch{i — u sec //900) =: v^h. 

 For latitude 60° and u = 20 the value of u sec //900 is abou t 

 ■04i so that we see for ordinary speeds and ordinary latitudes 

 reached the term involving u may be neglected, and the 

 equation reduces to 



- 2C/( = z; If 5 (3) 



For a star 



dVl _ length of mean solar day _ 

 dt length of sidereal day 



with sufficient accuracy, and as before we obtain equation (3). 

 For the Moon 



dH _ length of mean solar day _ I 

 dt length of lunar day I '035 



and 



U = 869, 

 so that in this case (2) becomes 



- 2C/i = 1-035 (J'T*)- 

 On giving C its proper value, and putting :</ for the velocity 

 compounded of the velocity in latitud; of the ship anl t'le 

 velocity in declination of the body (3) becomes 



h = ~ zw(tan / :f tan S) sin i' cosec- 15' . . (5) 

 NO. 1325, VOL. 51] 



The reduction is therefore from (i) 



X = - '—(tan / :f tan 5) sin 1 " cosec- 15' . . (6) 



Equations (5) and (6) therefore give the hour angle (from the 

 ship's meridian) and reduction for the Sun or a star at the 

 maximum altitude. For the Moon I'oiyui must be substituted 

 for w. The form of these equations has suggested the construc- 

 tion of the accompanying table, which gives the value of 



tan x° sin i" cosec" 15' as far as -r = 60. Thus in latitude f 

 when a body (Sun or star) of declination 5° is at its maximum 

 altitude, the sum or difference (according as / and 5 are of 

 different or of the same name) of the arguments corresponding 

 to /and S multiplied by w gives the hour-angle in minutes of 



time : the sum or difference multiplied by ~ gives the reduc- 

 tion in seconds of arc. For the hour angle of the Moon 

 the sum or difference must be multiplied by i '035tf, and for the 

 reduction by i(f035w)-. 



Example. — D.R. latitude 48^ N. Moon's declinaaon iS' 48' 

 S. decreasing 120 per lom., ship steaming S. 20° E. 16 knots. 



Remembering that when the ship is in N. latitude, and 

 steaming towards south 



w = - z; :p 5, 

 we have 



and 



w = - 27, 



/« = 27 X i'035(-283 -1- -087) = io-3m 

 X = 144" 



If a ship whose maximum speed is 20 knots does njt reach a 

 higher latitude than 60, the greatest values that /; and .v can 

 have, are for the Sun, 9m. 15;. and l' 37 ; for the Mjon, 

 17m. 20;. and 5' 41 ". 



I will now investigate the nature of a diagram ' from which 

 the reduction in ordinary ex-meridian observations may be 

 obtained, as well as the hour angle and reduc'-ion for o'.aximum 

 altitudes. It is found convenient for this purpose to express x 

 in minutes of arc, when (i) becomes 



/(- = 2-v(tan / If tan 5) sin i' cosec -15' ... (7) 



Now if a circle of radius a be referred to a point in its cir- 

 cumference as origin, its equation is 

 r^ = 2ax, 

 r being the radius vector and x the abscissa measured along the 

 diameter through the origin. Comparing this with (7) we see 

 that if asystem of circles bedescribid pissing through a common 

 origin o, their centres being collinear and on the same side of 

 o, the reduction > is the abscissa of the point whose radius 

 vector is /; on one of these circles: the particular circle on 

 which the point lies being that whose radius is the sum or 

 difference of the ordinates of the curve 



corresponding to 



y = tan x sin I ' cosec- 1 5 

 .V = / and X = d. 



1 A diagram based on the properties of the parab jla wa? given by ProC 

 Foscolo, of Venice, and publtahed by the Hydr^graphic OfSct about 1870. 



