We na ( tte? Om ape —_ ~ eon nes Oe a Ve 4 ret “Se chan! Noe) ae HERE ance a ors coon aS ae ae 1, reer y, per eS de Ce | Oey v. Gy ne Nes ROS ne a eh a . tae ey en +b av a hd ee : = , a. ; y +: b 5 Me nfo a $ «= 2 eae ¢ Pre DR tnt of, wat i 3 eg ads # . oe .4 ‘ a “ : a . » 2 p ‘a , ee ont vAG Aa Rete. - PROPRRTY — A Civil Engineering Departaieull . | UNIVERSITY | OF ILLINOIS, a “ “s 4 Please return to ) ENGINEERING HALL cA a ae Room No: 22-2 Gase No, se Drawer No. Drawing’ Wo, sccm a hs : REMOTE STORAGE © Property of the . oN eis DEPARTMENT uy THE UNIVERSITY OF ILLINOIS LIBRARY ay eae, ole Wn ATE »? UNITED STATES HYDROGRAPHIC OFFICE. No. 91. dae 3.1 OF MERIDIONAL PARTS, TERRESTRIAL SPHEROID, i oe Ux ee Compt BOT ie ae BY G. W. LITTLEHALES awnp J. 8. SIEBERT, Ur 5- FY DROGRAPHIC OFFICE: Under the direction of LIEUTENANT G. L. DYER, U. S. N., Hydrographer. WASHINGTON: GOVERNMENT PRINTING OFFICE. 1889. - Vf pe. ‘ 7 . 4°34 ay <3 ) “ 7 ri : ‘ , 7 r . + Pint < * » 5 , ? , ve - - . ’ j 7 ® : ° ‘ . 7 < - # a” . ? / . . x! > te ’ ' * ‘ - x é 2 | add BORE ath Cy ne Se oe INTRODUCTION. The Mercator projection (named after its originator, Gerhard Kra- mer, a German, who, according to the custom of that time, 1569, Latin- ized his name) results from the development, upon a plane, of a cylinder, tangent to the terrestrial spheroid along the equator, upon which the parallels of latitude and meridians of longitude have been projected in such a manner that a loxodromic curve, or the track of a ship when the course is invariable, shall appear upon the development as a straight line. The line of tangency, or the equator of the earth, being the circum- ference of a right section of the cylinder, will appear as a straight line upon the development, while the different elements of the cylinder, cor- responding to the series of terrestrial meridians, will appear upon the development as a system of equidistant straight lines, parallel to one another and perpendicular to the equator, maintaining the same relative positions as the corresponding meridians have on the terrestrial spheroid. The series of terrestrial parallels will also appear as a system of straight lines, paralle] to one another and to the equator, and will so intersect the meridian lines as to form a series of rectangles whose successive heights must increase in proceeding from the equator in such a manner that the required equality of angles shall be preserved for correspond- ing elements of the loxodromic curve on the sphere and of its repre- sentation on the development. The computation of a table of Meridional Parts, or Increased Lati- tudes, has for its object to determine the respective distances from the equator of the various parallels of latitude, in accordance with the ascertained law of progression by which the fundamental condition of the Mercator projection shall be realized. FORMULA FOR COMPUTING MERIDIONAL PARTS. The figure of the earth is very nearly that of an oblate spheroid, that is, an ellipsoid generated by the revolution of an ellipse about its minor axis. The section made by a plane through the earth’s axis is nearly an ellipse, of which the major axis.is the equatorial and the minor axis the polar diameter of the earth. Accurate geodetic measurements have 3 248735 a Shown that there are small deviations from the regular ellipsoid, but it is sufficient to assume all the meridians to be ellipses with the mean dimensions deduced from all the measures made in various parts of — the earth. Referring tothe figures, let Pm’‘cte’P’e’’ represent a meridian of the terrestrial spheroid whose polar axis is PP’, and whose equator is ee'e/’; let darepresent an element of a loxodromic curve upon the sur- face.of the sea between two consecutive meridians me and m/e’; and let LO represent the same element on the Mercator chart, as included M' mM ¢ T E' E between the corresponding meridians ME and M’R’. Let lé and LT be taken parallel to the elements ee’ and EE’ of the respective equators included between the same meridians. Then the condition before men- tioned, that the angles of bearing lct and LCT shall be equal, re- quires that GD sch sleet or, since LT=EE’=ee/=the element of the terrestrial equator, it is necessary that OT :.ct :: ee : lt Hence putting dm=ct, the meridional element of the terrestrial spheroid, adM=CT, the meridional element of the Mercator projection, a=the equatorial radius of the earth, and r=the radius of the parallel of latitude of which /¢ is an element, | we have, since the elemental ares ee’ and It are proportional to their respective radii a and 7, aM ee a dm lt a ° . . r e e e e e (1) as the fundamental equation, expressing the law of the Mercator projection. | 5 At any point on the spheroid, let p=the astronomical latitude, fP=the radius of curvature of the meridian section, p, =the radius of curvature of the section normal to the meridian, c=the compression of the earth, and e=the eccentricity of the meridian. By compression is meant the difference between the equatorial and polar radii, expressed in parts of the equatorial radius as unity, or cL a in which D=the polar radius. The eccentricity of the meridian is the distance of either focus from the centre, also expressed in parts of the equatorial radius as unity, or ea ati) 2 a e=(2c—c’)? The values of the principal radii of curvature are— pP,=a (1—e’) (l—e? sin?p) 2 and a p,=a (1—e’ sin? ~) ? Observing the figure, we have, as the length of ct, the length of an element of the meridian, se ff ai tere te Oe A Ce aE | 4, and, from equation (1), dM, the corresponding length of the chart me- ridian = - xdm; Therefore, dM=" . p, . dy ch secant Lally (3) From the figure, V=/,-COS P=AaCcOS Pp (1l—e’ sin? p) ? eer came oe, , (£) Substituting in (3), the value of pm, and the value of r from (4), we have _— al—é)dp . (arena cout etree) fay 2) Equation (5), by integration between the proper limits of latitude, will give the required length of any finite portion of the chart meridian, representirg the corresponding meridional arc on the terrestrial sphe- roid within the same limits. 6 For this integration, multiplying e’ in the numerator by sin? p+ cos? g, equation (5) may take the form dma“: 2? _v cos p. dp cos pm 1—e sin? p Again, multiplying the numerator and denominator of the first term by cos g, substituting 1—sin® ¢ for cos’ ~, and resolving both terms into partial fractions, the last expression becomes _a/( COS Pp COS P_\ 7,4? ( _€ COS P € COS M ae 1+sin mg 1—sin p ae sin @ Ta e sin ws ai For the length of that part of the meridian of the Mercator chart between the equator and any latitude, mg, we have we Mee COS @ COS Pp ) re € COS ~P € COS @ pee (ryan gt gp 1—sin p “s \Ge sin pp Bt ain P ae a LAE Sp ae rie sin @ =«(5 log l—sin pg 2 eo obee een Pp in which the logarithms belong to the hyperbolic system. Converting the logarithms in the foregoing expression to the common system, it becomes M=the meridional parts=a log, 10 log 5 I+sin p_1 ojo¢1+¢ sin p —sn gp 2 1l—e sin p Substituting for ma aie its trigonometrical equivalent, and for I+¢ SID @ its algebraic development, we obtain 1—e sin Pp M=a log, 10 log tan( 459+ f )—a( esi g +56 sin® ptre sin? p+. -) which is the formula for computing the values given in the following table: COMPUTATION OF THE TABLE OF MERIDIONAL PARTS. The following numerical values, with their respective logarithms, have been employed for the several constants: 1. The equatorial radius is expressed in minutes of are, in order that the unit of measure of the roeridional part M may become a minute of the equator, or a geographical mile. The value of a is, therefore, __360° x 60! 10800! rl b he SAG ENIRRGTR CS ee - + « « ». lOS== Seems 2. The value of log, 10 is 2.30258509 . . . . log=0.3622157. 7 3. The eccentricity of the meridian is that based upon Clarke’s deter. mination of the compression in 1880, which is C= 5p gg 0.003407562 . . + + log=7,5824437, whence we have e=(2c—@)*=.08248146 . . . . log=8,9163666 Hence the values of the several coefficients with their logarithms are a log. 10—7915'.7044558 . . . . log=3.8984895 Ge ==) ano 11. oh eas LOS OG900T2 4ae¢@ == 8 0’.053042 ees NLOP=RO 246192 tae = £0/,000216523 . . . . log=6.3355038 Accordingly the formula for practical use becomes M=7915.7044558 log tan (45°+ 5) — (23/.38871 sin p+0.053042 sin? p+0.000216523 sin? p+ . . . .) The computations have been made with this formula, aided by inter- polations, to four places of decimals, and checked throughout by first and second differences. 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