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'->«- -; -^/ 3 «-» v*^ Pf TH Pro/esso: A COURSE — OF — Practical Astronomy FOR SURVEYORS —WITH- THE ELEMENTS OF GEODESY —BY- LlEUT.-COLONEL J, R, OLIVER, R,A, d^^d cmKi^cr Professor of Surveying at the Royal Military College of Canada. mm. KINGSTON : PRINTED AT THE DAILY NEWS OFFICE. 1883. ' > ■(yc J J V.1 A Thi< Cadets first fi^ portion this CO! ters, to, Geodes ditional Governi TopogfE necessar cause, w themselv course, ti had to re would ha to make diagrams tended to ments et c also made ^.^ IN PREFACE, This manual has been drawn ud for th Cadets of the Royal Military Col. rl "'" °^ ^^' first fivo rKo . ^»"tary College of Canada. The nrst nve chapters on PrarhVa I a.* portion of .^e .u.Je« w^h t/, TuT, ^^-^ ">a. this county- ought to be familiar Th. '""" '" necessary to draw un ^''°'"" absolutely f to draw up some compilation of this UnH u diagrams has hTen ^ IT.:?"^''."'^ '"^ —"- of tended to supply .he^l r """"'"' " ''^'"? in- dents ...r^tteli^srSr ^r;^ 1 'T- ---the higher portion of j^ro;:?:;:: ."^ e^ 33^72'B' IV Preface. as brief as possible. It will be found treated in the fullest manner in Chauvenet's Astronomy. Geodesy being both a difficult and a very extensive sub- ject no attempt has been made to write anything like a treatise on it. All that has been aimed at has been to give a sketchy account of its most salient points, adding a few details here and there. The student who wishes to pursue the subject further is referred to standard works, such as Clarke's Geodesy. The author has to acknowledge having made more or less use of the following: Chauvenet's Astronomy, Puissant's G^oddsie, Clarke's Geodesy, Frome's Trigonometrical Surveying, Loomis' Practical Astronomy, Gillespie's Higher Surveying, Deville's Examples of Astronomic and Geodetic Calcula- tions, the U. S. Naval Text Book on Surveying, and Jeffers' Nautical Surveying. He has also to thank Lieut.- Colonel Kensington, R.A., for valuable assistance in in- vestigating some doubtful formulas. Kingston, Cadada,| January, 1883. j i CONTENTS. PART I. PRACTICAL ASTRONOMY. General view of the universe. CHAPTER I. ,r,o«„;...^ J J- . '^Ijf, ^^^^ ^*^''^- "T'^eii- classification, magnitudes and distances. The sun. The planets. Their rela- tive sizes and distances from the sun. Apparent motions of the heavenly bodies. Their real motions. Motion of the earth with reference to the sun. The solar and sidereal day Mean apparent solar time. The equation of time. Sidereal The sidereal clock CHAPTER n. and time. 'meridian," •latitude," "longi- , -ui L . " ' — ..>ude," "azimuth " sensible horizon "rational horizon," "parallels of latitude " dechnation parallels" "circumpolar star," "transit," "paral- rll.f ^^[T"°"t„ ^^^ ^^"^1'=^' Almanac. Sidereal time.^ The celestial globe. Illustration of the different co-ordinates on the great sphere CHAPTER III. Uses of practical astronomy to the surveyor. Instruments employed in the field Their particular uses. Corrections to be applied to an observed altitude. Cause of the equation of time. Given the sidereal time at a certain instant to find the mean time To find the mean time at which a given star will be on the meridian Given the local mean time at any instant to lind the sidereal time' I lustrations of sidereal time. To find the hour angle of a civen star at a given meridian. To find the mean time by equal alti- tudes of a fixed star, To find the local mean time by an observed altitude of a heavenly body. To find the time by a meridian transit of a heavenly body _ _ 22 VI Contents. CHAPTER IV. To find the latitude by the meridian altitude of the sun or a star 1 he longitude. Differences of longitude measured by differences ot local time. The meridian. To find the azimuth of a heavenly body from Its observed altitude. To find the meridian by equal altitudes of a star. To find the meridian by the greatest elonga- tion of a circumpolar star. To find the meridian by observations ol high and low stars. Azimuth by observations of the star at any hour CHAPTER V. Sun dials. Horizontal dials. Vertical dials pole 36 49 53 66 CHAPTER VI, The Refracting Telescope. The Micrometer. The Reading Micro- scope. The Spirit Level. The Chronometer. The Electro- Chronograph. The Sextant. The Simple Reflecting Circle The Repeating Reflecting Circle. The Prismatic Reflecting CHAPTER VII. The portable Transit Telescope. Its uses and adjustments. Methods of correcting the meridian line. Effect of inequality of pivots. To apply the level correction to an observation. To find the latitude by transits of stars across the prime vertical. Effect of an error of deviation on the latitude. The pernonal equation.. CHAPTER VIII. The Zenith Telescope. Its use in finding the latitude. To find the corrected latitude. To find the level correction. Value of a division of the level. Value of a revolution of the micrometer screw, Reduction to the meridian. The portable transit instru- ment as a zenith telescope. CHAPTER IX. Additional,, methods of finding the latitude.— By a single altitude taken at a known time. By observations of the pole star out of the meridian. By circum-meridian altitudes CHAPTER X. Interpolation by second differences. Examples. To find the Green- wich mean time corresponding to a given right ascension of the moou on a given day. Interpolation by differences of any order. To find the longitude by moon-culminating stars. To find the longitude by lunar distances 89 CHAPTER XI. To find the amplitude and hour angle of a given heavenly body when on the horizon. To find the equatorial horizontal parallax of p heavenly body at a given distance from the centre of the earth. To find the parallax in altitude, the earth being regarded as a sphere. Star catalogues. Differential variations of co-ordinates. Correction for small inequalities in the altitudes when finding the time by equal altitudes. Effect of errors in the data upon the time computed from an altitude. Effect of errors of zenith distance, declination, and time upon the latitude found by cir- cum-meridian jiltitudes. The probable error loi 77 84 Contents. VII PART II. ^■-p # III GEODESY. CHAPTER I gvenlat ude Tn fin^ *k^ length of the great normal for a CHAPTER II Geodetical operations. Methods adopted for maoDinB countrv t,; '_, .. CHAPTER III. CHAPTER IV. sphere described with radius equa^Rennrmfi if Jl ""!?'"^-7 Reduction of a difference of laJi^nnAo *^ u' °^-^]^^ spheroid, responding differencr of Lff H. "i • ^ ^P^^''"''^ '° ^^^ cor- N„n,eHcaU„„p,es,''',5JS|?|LV'S;r/SiS'.5^S li 133 vrn Oontents. North 145 or's, CHAPTER VI. Trigonometrical levellincr t« c j . another, Reciprocal obrervatfonsfor^"'^'^' f!- °"= ^'^tion above ?or tr '° '^" ^"'"'"i' of i^ef^Jt cf.f'J^^ refraction. Re! by a single zenffh ^;J: '_"^ ^^ feciprocal zenith dis»an..c "-j 166 ^y. a single zenith distance N.Tm^''"'' , ^^"'^'^ distances, and height of a station by the zenith H,^'"'^' example. To find the find the co-efficient of terrestriSr^'/'^'!^^ °l '^^ ^^ horizon To tions of zenith distances! '^^acfon by reciprocal observa? The use of the pendulum ^^'^H^^^- >""• earth. The eLct o the "herofdr??'"^ *^ compression of the I ^.°l^"i-'- obse^ieforSta" Tf/J!:«-.l-.^ 'he ffo^f 175 the pendulum must he ti^ ' ^Z— — ' <^ in ord^er"?hL^-»"'"'^'" ^^^' *»>« length of obser- 182 5- Deville's nd the area *o parallels a parallel 145 Projections. Mercator's, 166 tion above :tion. Re- nuloe used the differ- inces, and find the izon. To 1 observa- 175 an of the the force a pendu- e time of length of a given centre of Kater's > obser- 182 NOTE TO PAGE 52. By drawing a figure it can be easily shown that, in the case of a horizontal dial, iff is the latitude, P the hour angle, and a the angle the corresponding hour line makes with the meridian line, then : sin f =cot P tan a or tan a=sin ip tan P. Similarly, in the case of a dial on a vertical wall facing south, tan a==cos tp tan P. In the latter case the angle « is measured from a ver- tical line on the wall. The stile is, of course, set parallel to the polar axis. We can thus find the hour lines for each hour, for any given latitude, by solving these equations. C tj^"ad fron, thoir surhce, am • 7 "^."'^ »""'«'" ^Aected ■elascope, look lie sma I J """"' "'™''«'' '^ KO"" noticed aiso to pasHl™ Xef IL^hI™^ "= ^'-^^^^■or— ?IS^^^^^^^ .0 con. anrnTZlrn^tti^rwttftl-r^^"''™'^^^^ ^tirandVel^i^is'^S-r'"'-^'^"^^^^^^^^^^^^^ When the moon is first ^pph ^o ^ little to the east of the sun sh^ ^T^ '"°°" ''^^ ^' ^ or tne sun. She rapidly moves through i 6 Tne Moon. the sky towards th- east, so that about full moon she rises as the sun sets, and later on is seen as a crescent rising before the sun in the early morninj,'. The heifjht to which she rises in the sky will l)e observed to be (unlike the case of the sun) quite independent of the time of year. The interval between two new moons — th a t is t he ■ time s he tak e s t o t H ftW^^-^ppareftt ciretij^ of the-sky — is about a^'days ; and she rises eaci'^ilaTati^t^irJo .|u^^ -t ern - of an hou r later than the day before. The stars, if carefully observed, will be noticed to" rise each night a little less than four minutes earlier than they did the night before, so that at any given hour a certain potion of the sky which was visible at the same hour the EBifeiit before will have disappeared in the west, and a similar portion will have come into view in the east. In fact the whole mass of the stars appears to be slowly over- taking the sun (or rather the sun to be moving through the stars); and, as a consequence, if the stars were visible in the day time this motion could be plainly seen. The points of rising and setting of the stars are always the same. The sun and all the stars reach their greatest height in the sky— or culminate, as it is termed — at a point where they are due north or south of the spectator. The stars in the northern portion of the sky, from the horizon up to a certain point depending on tli-^ position of the observer, never rise or set, but describe in the twenty-four hours concentric circles round an imaginary point called the pole, and in a direction contrary to that of the hands of a watch. The different p w.n, if carefully observed, will be noticed, not only t > > it, ge their positions among the fixed stars., but to \rsy i };:ivhtness from time to time. So much for the upp..-ent motiohs of the heavenly bodies. We have now to consider their real ones. J The Earth's Motion. The earth describes an eUiptic orbit round the sun in about 365^ days. Ii also revolves on its own axis in about a day. Thi': a\n roniains parallel to itself and is in- dined tf. (h ■ plane '1 the orbit at an anf,'lc of about 2{" 37'. Hence the phenomena of the seasons, and of tlie varyin;,' positions of the sun from day to day. Figure i shows the position of the earth with reference to the sun at the different seasons. N is the north pole, S the south pole, and A a point in the northern hemis- phere. The left hand sphere shows the earth's position when It is midwinter at A, and the right hand sphere when it is midsummer. The motion of the earth round the sun causes the latter to continually change its apparent position amongst the stars. Its path through them is called the eciiptic, and hes, of course, in the plane of the earth's orbit. The earth's revolution round its own axis, although on an aveiage 24 hours if taken with reference to the sun, really tako. place in space in about 3 minutes and 56 seconds less than 24 hours, the difference being, in fact, the same as that between two successive risings or settings of the same star. It should also be noted that, owing to the enormous distances of the fixed tars from us, all lines drawn from the earth, no matter what its position, to any star, are sensibly parallel. " culminating a er tlthFlV ^"""^/ ™-ta™ of I4i minutes abfut he lltn February, and then continues to decrease becom n^ ^ero agam about the Z4th of April. ,t attains a marimZ o. 3m 50S about 14th May, becomes zero about wth The';!* ""7.f '■"'"' ^'"^ ■'"'^' ™d -- 31st Aup^t The cause of the equation of time is as follows If h ' rate and if the ax,s on which it itself turns were peroen each d,.y at noon exactly. But the earth moves in an ellipse and at a variable rate, and its axis is inclined to the plane of the ecliptic at a considerable a„gle!th" com bined effect bemg that we have the equation of lime. iJ*'\^^^ ""''^ "" "''= '^^rt'' ^hose plane passes a rX '""'""""'= " "»■" »"Sl^= to the axl s called the "equator," and the projection of its plane ,n the .noved at a unifortraTlnTtrsTdTe'eq^Tn-octirrL'' tead of the ecliptic we should have no equatln oHm"' «n imaginary sun moving in this way is called 1"™™' .hit" tlpl't" Ih *^ '™' '=^P' ''y » "''""y clock and nat kept by the sun-in other words "mean time" and "sideSr im:"'.h'T'"~r "^^^ ^ """^ «-" - ' sidereal time, that .s, the time kept by the star., I, has been already mentioned that the interval betw^n tjo successive culminations of the same star is aMe tes than .4 hours , the time it takes, in fact, for th ar.h 10 Sidereal time. to make a single revolution on its axis. If we divide this interval into 34 equal parts we have 24 sidereal hours; and f we construct a clock with its hours numbered up 0/4 instead of 12 and rate it to keep time with the stars, it ts easy to see that the hour it shows at any instant wil g ve the exact position of the stars in their apparent diuf^al revolution round the earth. Clocks and 'hLometro thisdescnptionareused-theformerinfixedobservatories. the latter for surveying purposes. The subject of sidereal time will be referred to later on Before proceeding further it will be necessary to explain the meaning of the various astronomical terms in ordinary li iivide this lours; and up to 24 stars, it is t will give nt diurnal meters of rvatories, ' later on. explain ordinary CHAPTER II. EXPL'. ,AT10N OF CERTAIN ASTRONOMICAL TERMS. THE NAUTICAL ALMANAC. For practical purposes the earth may be considered as a stationary globe situated at the centre of a vast transparent sphere at an infinite distance to which are attached the hxed stars, and which revolves round it in a little less than 24 hours. The sun, moon, and planets appear to move on the surface of this great sphere, the sun in the ecliptic the rest m their respective orbits. ' The extremities of the earth's axis are called the poles • and the poles of the great sphere are the points where the axis produced meets it. ^^ Great circles passing through the poles are called meridians." This term applies both to the earth and ! n Tf/Z^v^'^: ^" *^' "^'^ °^ *^^ ^^"^^ they are also called decimation circles." Meridians are also called hour circles." and the angle contained between the planes of any two meridians is called an "hour angle " be- cause It is a measure of the time the sphere takes to revolve through that angle. It follows that the hour angle is the angle formed by two meridians at the poles. In speaking of the meridian of a place we mean the great circle passing through the place and the poles; and a great circle passing through the poles of the great sphere and the zenith (or point in the sky immediately I it 12 M Latitude and longitude. over the observers ht^:>A\ ic ♦!, . ,~ as regards the/raa.:;!':."' '"""'''" '°' ""= '-'-'• "longitude " The fn ^"-""^T"'' '^''"«' "'"tilude" and poinffroltheluatniid'' ""*^'''" *'^'^"" "' ->' north or south as the V» "'T'"''^ "'""^ " ">"««" -es.o..ro:tt:ru:.rr;^artL';r"- Piat:r,:;t ce r'tarrj" V?"" "' '"^ "«■■■''- °f 'h^ either b, .L "; e,: ^'orth'''''"' "'" '^ "'^^="-' angle contained by 'the two 1 H """"°'' "' ^^ ""^ measured east for z8o»M , '"""'"'a"^- Longitude is countries reclcon from diff" T'- -^T "*°°- J^'^^^"' English use that o GrUt d, T,""" "'"■'*^"^- ^'- many inconveniences !nr. T P''"^"' ^J"^'"" ''as the world wiuTnite 'n Jd^ V'' ''' '"'P'^'' "'^' ^°™e dav Will reckon lon^i::!" h ^"the'wToll",! T'''^" '"'' Stead of as at present. "^ ° ^^^^'^^^ "^- is Itr^n^ted Vltalla^r:!;'"^ ^= ?" ">^''-' 'Pl^-e called "declinaL ™ and "ighTa ' '■" "" '^"^^ »- corresponding .„ latitude nd t rCeT' ' f '°™^'- Decimation is measured fr^.^ ,1, '° '"ng'tude. •he poles, and right rctnreasLrr"^' '"^^''^ meridian. The latter ;= h^ ^astwaid from a certain Whole 36o",and isttn^dXhTu:;':, ^ '"^7^" '"" .nst«.d of by degrees, t houf corrrsp^^nX^d" The point where the 2ern or ^. i • equator is called the ''Cpol t TrLs"^^''"; "^^ ''' nated by the symbol Z'. It isalsoot f .^ "^ " ^'''^'■ of theequatorwith theeclintir n . *''" "'tersections -wconicaimUrtte:r^h.:tis^::i;:te^^^ Altitude and azimuth. are no. ..e,„.ed J^ Xr^ he^.^ dtf ^' '"' to fttre?a*rXro";'°"'? =°-"'"""- *'■-'' -late is necessarv to^vT ?°'"" °" " 'P''"^ """'her set -fere,: r^ hel t'j;!^" "^ ^ "eavenly body with "altitude" and "a. " th " rl 'f '"'• '""=>' "^ -^^"'d Planation. The se Idt ,h J f"r' """'"' ""''^ ^''- cal plane passer h"'h 'he "t T' '/ '"' "^"'- with the plane ofthe obfe„er^ ™ t dtn "th"' ,°''^^" and azimuth of a star=.t,„! "^"""n. The altitude -d ,y the v^.^iotrirn'r^r :n' :!;^ tit star Azimuth f "''" ""^ ''='=" "'i'-^ted on the -nd4r:L;:soT:s rs^it ir "■= -"^ reckoned from the south. '' sometimes The plane of the "sensible horizon" is th^ h • plane passing thrniicrh fu^ u , is the honzonta fore ta'senS teh" ea,.ht"7 ' """u"™' ''"^ '"ere. "rational horizon" is an In, n'^ " "'" P"'"'' The sible horizon and olinJ H ''"u ''.' '" """ "^ "■« ^'^°- TI>e ProiectionfofT ^'.loZtf :„7h " "' "^ ^""■• coincide, beinsat an inhnUe dl^nce '''" ^P''"'' si.ht 'itZy tZt'^t^"' ^'^ '^ ^-' ^Ph-e is in generally on thettitLe 71°" ^ " "^""= "'P^'"'^ time of the instant At the n -I P'"," ''"'' ""^ ^'^ereal - hemisphere wUVe-^-' P^^ a^dtoX i ' 3 14 Definitions. ! ;.;: . \ tf Lu h.r^V • K P°'' "■" "'"' ^-M •>= limited to the beo^theh""' "•- '^' '"^ '=''"="°'- "<>"■ P<»^3 would be on the honzon, and every point on the great sphere would come m sight in succession. At intermediate pkces a certain portion round one pole would always be above the horrzon while another portion round the o her pole would never be visible. "Parallels of latitude" are small circles made by the in- equator. Similar circles on the great sphere are called 'decHn tion parallels." A little consideration wHl show that withn a certam distance of the equator at each side at mid d '""rrK /""' '" ^^^ y^^'' P^^^ overhead at mid-day. The belt enclosed between the two parallels withm which this takes place is known as the ^tropics." A few more technical terms require explanation. When peaking of the "hour angle" of a heavenly body arany instant we mean the angle formed at che pole by the meridian circle of the instant and the dedination circle passing through the body. ^ By the term "circumpolar star" is meant a star which r "Te ThTT '^ ^--^-on^PletecircJound he pole. These stars cross the mei^dian twice in the twenty our hours. One crossing is^called the "upper ransit • the other the "lower transit." At the points be twcenth. transits at which the stars have the greatest azimuth from the meridian they are said to be !t the greatest elongation," either east or west. The words "transit" and "culminate" have the same meaning when used with reference to stars which rL and tio7oTnr ''i '' '^" ''''"^' ^" '^' ^PP^^^"t relative posi- t on of objects owing to a change in the observer's posi- u,n^ Astronomically it generally signifies the difference m the apparent position of a heavenly body as seen by an ted to the es would It sphere srmediate Iways be the other >y the in- lel to the "e called ill show ich side >verhead parallels ropics." . When at any by the n circle ■ which e round in the "upper ints be- reatest t their ; same ise and ! posi- i posi- ;rence by an Parallax. ^ ^ ot the earth. Parallax ,s greatest when the object is on par: ikx Th ;"'f T' ^'^ ''''''' '^^ ^ considerable cal Almanac are those which they would have as seen oTe ctairf'""^"' ^"' '' '^ ''^'^'-^ necelryTo correct all observations on those bodies for parallax. .^fl^'f'^\^T"" *^' °^j'^' *° have less than its true altuudc Refraction has the opposite effect. The latter like the former, diminishes with the altitude. Near the hon.on-say within xo degrees of it-its effect is very un! certain, and observations of objects in that position are s altrr' "a"''" -^^ ^" ^^^'^"^^ ^^45° the' refraction IS about 1 . As It vanes with the temperature and atmos- pheric pressure the barometer and the thermometer must be read if very exact results are required. The corrections for refraction and parallax are not to be found m the Nautical Almanac, but are given in all ets of mathematical tables. The N A i. n mil , variable quantities - such as' tdLLt^lircet sion, equation of time, etc. It is rather a bulkv volume but he portions of it in general use by the practical survLyo could be comprised m a small pamphlet. The most use- ful are the sun's declination and right ascension, the equa- t on of time, the sun's semi-diameter, and the sidereal time of mean noon-all given for every day in the year ; the de ,,„ tions and right ascensions of the principa fixed stars, taken in regular order according to their rieht tTmTiT'^r' ;'' *''^" ^"^^°"^^^*'"^'"t--J^of mean TfllV.'^^^ ^'""^ ^"^ ''''' '"''^- To these may be added tables of moon-culminating stars, and tables for oSthTmtidtr ^ ^^°^ ''' '''''' °^ ''- ^^'^ -- -hen 1 ■ m i6 The Nautical Almanac. JeltTT^ "^ '^k'- *"° ^''' P^^^^ °^ the quantities t.on for any other hour or bne^tude T. ^' ^ ^'^P^'" change in six hours. corrected for their and the sidereal time i,h.. they would bio al ' ^'"'C respectively a, a place in longit'ude go- wes't "' =''• In the Nautical Almanac the day is suDDo^eH , -^^:;:/:;fe:xiorT^ttetr;- Iati„g. ^- ^""^ '^ f°^ <=™™ience in in.erpo. ' quantities , and of the the quanti- ty in ques- a propor- when it is -irswest of fore, if an it noon the I for their 'ti its axis id sidereal s 3 p.m., ■ and 5h. to com- ) a.m. on fh. of the sckoning 5aJ time, also at 32 days ig really interpo- . Nautical Almanac, 17 JUNE, 1880. AT APPARENT NOON. THE SUN'S Tues. Wed. Thur. Frid. Sat. Sun. Mon. Tues. Wed. Thur. Frid. Sat. Sun. Mon. Tues. h m s I s ^4 39 071 10-241 2 [4 43 671 10258 3 i4 47 1310 10274 4|4 51 19-87 lo' 289 514 55 2699 10-304 64 59 34'44|io-3i7 7 '5 3 4220 110-329 8 j5 7 5024 10-340 91(5 II 58-54 10-351 Apparent Var. in 1 Declination, hour I Sidereal time of| the Semi ((diamflter passine I *^« Meridian 10 12 5 16 708 10-360 5 20 15-82 10-368 5 24 24-74 10-375 ! I 5 28 33-81 10-381 i 5 32 4302 110-386)' i5||5 36 52-34 |io-38o 13 14 N.22 8 59-6 19-79 22 16 429 18-82 22 24 2-9 17-85 22 30 59-4 16-86 22 37 32-3 15-87 22 43 41-314-88 22 49 26-5'i3-88'' 22 54 47-6 12-88 22 59 44-6|'ii-87 23 4 r7-3;io-86 23 8 25-7; 9-84 23 12 9-6) 882 23 15 2901 7-791:1 23 i8 237 6-77 l!i 23 20 53-8| 5-74 III Eqnation ofTimel to be *^t.^rom \ added to Var. Apparent] in 1 J>«W. I hour. m s Is 2 21-83 0.384 " 1242 0400 2-61 0-416 ^ 5243 0-431 I 41-90 0446 I 31-03 |o-459 f'72 1 19-86 !o-47i f75 I 8-41 0-483 879 o 56-70 [0.493 |:|2 044-75(0-502 °85 o 32-600-510 o 07 I O 2027 0-517 1-2-020-523 ° 4^0-528 o 17-35 |o-532 II ;[ ff =1 i8 Nautical Almanac. JUNE, 1880. AT MEAN NOON, I JO It §i o Tues. I 1 Wed. 2 Thur. 3 Frid. I 4 Sat. 5 Sun. 6 Mon. 7 I Tues. 8 I Wed. 9 I Thur. 10 li Frid. n Sat. 12 Sun. 13 Mon. 14 Tues. 115 Apparent KiKht Ascension, THE SUN'S Apparent \ Semi- Declination.! diameter, Elation of | Time, to be added to -'^iiht.ymi, Mean Time. h m s 4 39 III 43 709I Ij4 47 13-451 4 51 20'igl (4 55 27-28J 14 59 34701 P 3 42-431 15 7 50-44 j5 II 58-7i| j5 16 7-21 \5 20 1591 |5 24 24-80 \5 28 33-84 5 32 43-01 15 36 52-29 N-22 9 o-4| ,5 48-1 22 16 43-6 ,5 47.3 ^2 24 3-5 15 47.S 2^ 30 59-9 15 47-7 !i " 37 327 15 475 22 43 417 15 474 22 49 268 15 47-3 22 54 47-91 15 47-2 22 59 44-" 15 47-1 23 8 25 a 15 46-g 23 12 97! 15 46-8 23 15 290 23 18 237 ,23 20 53-9 15 467 15 467 15 46-6 Sidereal Time. h m s 4 41 2293 4 45 19-49 4 49 1605 4 53 i2-6o 4 57 916 5 I 572 5 5 228 5 8 58 83 5 12 55-39 5 16 51-95 5 20 48-51 5 24 45-07 5 28 41-6J 5 32 38'i8 5-.3li£74 APPARENT PLACES OF STARS, 1880 ^T UPPER TRANSIT AT GREENWICH. The Celestial Globe. 53 1 2 60 57 916 ' 572 12 'iZ l"t Tht tI """°" '' ""= ^'"f^^^"! "•"'« of the in. instan. o^.:: ^ , " te" v». .Tfi^aTh """ "°""^ ""> star, transit we have o7;t 'convt M^ Ta'^R" a' ■nto the corresponding mean time of the iL tant t '"'•''■'• ^'""'>' - the UUer. an. .e po^^ :: r ^^rrolT Explanation of Terms. 21 ea" :L?; A Zal "T;- '''"' '^"«'="' '° ""= observer's hori.l t^ „ p ''^"I" '"' '" ""•■ ?'=""= "f the great sphere in^h oinTs p p'^'""/,^' ■"-""« "■' "e plane of the paper represents the plane of f l,» observers meridian anri H tp .v ^ *"'^ points of th"Z2o^"\\r "" """'■ ^"'' ^°""' equinoctial, ; |" X' 1 oIi: I?'/''"''"', '^ ' ^ the »t . 1, ecl,nat,on circle passing .hrongl, i., fro/„h h f^^Tr' r "='°""'- ^ '= ^ ^'" -'-'ed on ", and s. another star on the observer's meridian 7 f^l' TIT "-^ "PP" the centre. If it is not ", if *" '" '"'''''''' '' ^till in P.ate screws, half byr'trir^'^r'aiifth^ '"^ '"''" repeated till the bubble remiinTTn ll "Peration position. The altitude TZt '„'''= "?'" '" -ery •s then taken, the telesc ope 24 A Ititudes. read. ^^^ ^°^^ verniers should be there is any. The followingTS ri,^e the c f™' "■ be applied in each case to an a tuda 'f 'hr > °"' '° or lower Ji™b to obtain that of h "centre "" ' """" THEODOLITE. Index error. Refraction. Parallax. Semi-diameter. SEXTANT. Altitude above water horizon. Index Error. Dip of Horizon. Refraction. Parallax. Semi-diameter. Double Altitude with artificial horizon. Index Error. Divison by 2. Refraction. Parallax. Semi-diameter. T^, — -^^'"'-uicimeter. is o^siTdiirr^tr v^;:„t^4f--;r such a position .Lt ey It^'a^H t^Vr^^" j"'° from each other At \°^r^f " '™ wh-'e receding observer cal,:*4';p,^ b assSn^ e^^r' ""'^'' "■' time, and the vernie; is then read Th "''" '""'='' observing the iower limbT e forenoon" tnTrt""''" m the afternoon Thp Hi,, a ""^^"^^^ ^"a the upper instrument above the Ltr 'd" iUthe ';'^'" °' '"^ Parai-a. is to he found in the^mJthemlt ^[rbt!" ^"', In the case of a meridian altitude for laHtn^o ,u or star, after rising to its greatest hd^hfpp^^^^^^^^ short time to move horizontally. When tins is th/ ^ the altitude may be read off ^''^ ""^^^ or'st1-rmrr:'^;s°?h?'SracrT' J '"^-^"- correction for temperatue and lr\ "'•''' ''''"■'^^ ^ be,ghtofthethermrt:rdtarmrx„';rrot:^! J i Equation of Time. _ ^ ^ 25 If an altitude has to be taken wi>h tho . " " artificial hori.on, and the sunt.oo h t t T h '"' a „,eridia„ observa^ilnTTr he ^ZTT.:''"^'' '"' Sji e: ratf- r ax" ""-"-^^^^^^ PH^^-C.P.x CAUSE OP THE EQUATION OK TIME. In Figure 4 P is the pole, E C a portion of the ecliptic, and E Q a portion of the equator; each being equal to 90°. C and Q are on the same meridian, and P Q is alsoaiuadrant. Now, let S be the sun, and suppose it to move at a uniform rate from E to C Let M.h: :: JaTaTirrr ^^-r '"» --'"^ .he .wo suns s.an to^lT.^'t.^^'Z; atTt''" interva let their nn<5.>mn u V ^ certain Since they move atT'Iear; E 5™-.!^""= ''^""• E S., hnt as a conscnencrthT^eridfanTp ^^^rp s" S P s" r'"':,''K' ':""'"« ^°' ''"ead of s The aLI i/n^ #--^n/--T_ '^>j«.*- H c-d-i--*- tf^c- :<^t<^ 'rf^ ■^^ ■e^tz.i.-^'j e'^^4. / ^^ 7' '■■O.'^t-rt^ ^y' f'. ^7^< ^ ^It-^l^J ^1 ^ /^ a K^i 'i^u.^.t^ ''yf ft ni,^^i/-i , ^j '?-fi,^-1^-*-ri-xj i;i : ^° Connexion of Sidereal particular instant and longitude, ^^ust allow for the change in the equation that has taken place since noon at Greenwich. For instance ; suppose we had to find the mean time corresponding to three hours p.m. apparent time on the 22nd April, 1882 at a place in longitude eh.' west By the N. A. the equation of time at apparent noon that day at Greenwich was im. 34s. 43. to be subtracted from apparent time and increasing, the variation per hour o.s.496. At 3 P.M. at the place it would be 9 p.m at Greenwich. 9 x os.496=4s.464. The corrected equation of time is im. 38S.89, and the true mean time 2h. s8m 2If3.II P.M. ^ GIVEN THE SIDEREAL TIME AT A CERTAIN INSTANT TO FIND THE MEAN TIME. Here we have given the right ascension of the declina- tion circle of the great sphere that is on the meridian at the ms ant, or-which is the same thing-the time that a sidereal clock would show. Now the Nautical Almanac gives tho sidereal time of mean noon at Greenwich, which nas CO be corrected for longitude. These two data give us the mtsrval in sidereal time that has elapsed since mean noon, and this, converted into mean time units will be the mean time. ' Ex. Find the mean time corresponding to 14 hours sidereal time at Kingston on the 28th April, 1882 We find from the N. A. Sidereal time of mean noon at Greenwich .u Con-ectionfor longitude. . "^eenwicn ah. 25m. 25S33 50 26 Sidereal time of mean noon at Kineston "T 7 Sidereal time of the in.stant "^'"^ston ^h. 26m. 158-59 14"- om. OS ""'^ircrm^^inrot.'.^''?!?'*'^^ ~ Which, converted into mean time' is "u " ^•^™' '*'^*'*i '^n- 3ini- 5OS75P. M. The conversion of sidereal into mean time units ~and 'I ' !■ I . «»<^ Mean Time. ^y Jl^^t f^'"^^- '''"' ""^ ""'^^ "°°" '^ grea^hlT^ sidereal time given we shall obtain the interval before mean noon. Thus, if on the same date as above we wanted to find the mean time corresponding to sidereal time one hour we should proceed as follows : Sidereal time of mean noon .. u < " of the instant 2h. 26m. ijseg I o o • Sidereal interval before mean noon 7"^^ 7~7" Which in mean time units is. . . T~ ' Subtracting this from i2h ^ ^° '56 12 o o We have mean time "T" 'Oh. 33m. 58S44 A.M. It is sometimes convenient to add 24 hours to the given suiereal time to make the subtraction possible. Thus.Tf the sidereal time were ih.. and the sidereal time of mean noon 23h., we should have the interval elapsed since mean noon 25-23, or 2 houis, which is ih. 59m. 40s ,Tm mean time. ^^ ^ ^^ '^'' TO FIND THE MEAN TIME AT WHICH A GIVEN STAR WILL BE ON THE MERIDIAN. This is only an application of the preceding problem IS the same thing as the sidereal time of its culmination and we have merely to find the mean time corresponding GIVEN TI... LOCAL MEAN TIME AT ANY INSTANT TO FIND THE SIDEREAL TIME. Here we must convert the interval in mean time that has elapsed since the preceding noon into sideraT units and add to it the sidereal time of mean noon. Apfil* iss'l'^ .?t"''r' 'J"^' "' 9 ^•"•' °" '^' ^9th of April, 1882, at Kingston, Canada. Here we have, as before : !!?7^l*'"® °i ""^^ "oon on the 28th. ... ,u « Add 2x hours of mean time in sidereal units.o;::::":::::,^- 3"' ^f g Sidereal time — 23h. 29m. 4as'67 pi lj F.n 'J*; "y , 28 Sidereal Time. h this process makes the result more than 24 hours hat number must, of course, be subtracted from it Thu f we got 35 hours the sidereal clock would show ih f the sidereal time of mean noon is greater than the inte - val m sidereal units we add 24 hours to the latter to make the subtraction possible. The correction on account of tongitude for the sidereil time of mean noon is constant for any particular pS The subject of sidereal time may be thus illustrated :" In Fig. 5 let the small circle represent the earth, and the large circle the equator of the great sphere viewed from the north, the plane of the paper m^^Em^t^m \ being the plane of the equator. IHHSB^HI J Let P be the pole, A a point on the earth's surface, and P A the meridian of A. j- is the first point of Aries, S and S» pi^ two stars, and . and s^ the points where their declin. tion circles meet the equator. Now the J/s^ t f^ rTs [Jt:?s' N^ ''''' ^^""^^°" ^' '^' -^ ^^e the meddian P /^^f^PP^-.t^e earth (and therefore tne meridian P A ^) to remain fixed, while the onf^r circle and stars revolve around it in the direction of tt arrow; and at the instant that it is mean noon on a c tain day at A let the position of the ereat .nh.,. y shown in the figure. The arc , s^ t ^TlteliL:^ time of n,ean noon for that day at A. The star S wH b on the meridian at an interval of sidereal time after mean noon corresponding to . n, while the star S^ ha pis ed the meridian by an interval corresponding tot/ and tZ 7^^r 'T'^'' ^° theirequivalnts n mean time we shall have the mean times of their transits. Fo i Hour Angle of a Star. ) was 2ih. jorn., the right as- cension of the star being taken «s ih. i5ni. Now the state ot things at noon would be as shown in Fig. 6. The star vvoiild have passed the meri- dian by an interva'l of 2ih. 30m.— Ih. i5ni., or 20h. 15m (sidereal) and would there- this the me „ ,41Z bf'L ''i'''.r"™'=- J^™" the contrary is the case ' '™'=' ='"" *==' '^ "' "'" ™' ""^' ™ - H8„.. .„„„,,3 „^ ^ I'lXED STAR. ^ tion of the alter A rr"'t ° "" ''""'^■' '" """"a- scribes a suffidently higr:,e1:".t'V''°r ""'='> «- houn before its culmin ,■ ^^- T"° °' three Its culmmatran «s altitude is taken with the it'll I 30 Mean Time by eqnal Altitudes. f ment still clamped) till irente s th fi.l^V"'^ ''"' telescope, and waits till t hasTxact v th. " - '^ '''" before, when he a^^.n n . !t, ^^ '^""^^^^'^^^^e as ic, vviien ne again notes the watch timp Tk» of the times of equal altitude will rive tiTwa, Jh r """, the star, cutaina.ion, which shouldbe theTarae T,^ mean time (previously calculateHi .„ ! "" star's right ascensio„^he la.t bei„7,7™^^ '° ">= of the culmination. f they Ire „o.,L '"'T" '™' ence will be the watch e,Tor ^' '™' '"' <"«f^'- come «owrf^ei"th: Slitit* Tbfhrontl ""'^ clamped and its slo^w motion screw used "' " TO ™b the LOCAI. mean time by an 'observed ALT! P„ .u- "^""^ O" -^ "EAVENLV BODY. ''™'""-"- for this problem we must know the l=,t,f.,j r i The altitude should be taken «,h«r, ♦%. l is rapidly rising or falling-thatTs L a%u^^^^ '°'^ about three hours from the mer dL and th^ " '' " the prime vertical the better. ^ "^^'^' *° If we take P as the pole, Z the zenith and S th. u venly body (Fig. 7). Pzg will be *^' ^'^- a spherical triangle in which PZ IS the complement of the latitude, PS the polar distance of the ob-' ject observed, and ZS the comple- ment of the altitude. The three sides being given we can find the three angles from the usual for- mulae. In the present instance we Fig 7 I Time by Altitude of Sun. want P. which is the hour angle of the object A convenient formula is Sin>— - 5iM£ziPS)^JnJi:iP^ sin PS sin PZ 31 wheS^ is ?^±PS+2S t?e\rtutt:J::°^Th'' ^",^^'^^^"* ^* ^'^^ ^-^^-^ to.J.o u. f ^"^ "sual corrections are apDlied s W^td^'r^i'ir"^'^ '''■^™^ beenworkeC """™ fy 15- This gives us the hour angle of th. o™ ™rs,t? r ^"'-""^ '!-= -„ecT.d : Lt" •he observa'ton ^ "" ""'"" "™ "'"« '■■^«""« "f If the body is a fixed star or planet then fr„,« ■. eX:r?d^d'ir^~ ="r' '" r -''^' '^' Insteacr of lakinj a single altitude two or three mav b, 3.':^ri==r,£;r.f.--S positions of the telescope and horizontal plate ^o as to correct instrumental errors c:h-1I ! ^ ^7tn April, 1882.— Lat. 44° iV 40" N T «« u 50s. W. Sun's semi-diameVer iV^V' 'n T' ^^' ^"'' 4o'o''N. Watch Hm.rrl^.^^' declination 10" Equation of Time Z tsTT^'l' ^'^ ''''' ^^^ -•«. ent time. Indr Cr'^;;!^, ^f!'*-*^' '^°"^ ^^P- ^ Example. Double altitude a.o ~. ~, Index error ^ '^ ° 5 30 ^^ Comt^Xc^/Ci^ 2)63 58 30 ■ y^ac A* ,4 a.!^ — ^''':fjr-:t^^Semi-diameter ^' 59 15 ^^^.^^:^. Refraction and parallax ... f 23 True altitude of sun'scer,tre^7~T7~^ 90 Declination Z^ f^TFlfl Latitude... 44 10 .^ „^ , 2l___f_4 91 26 15=5 A^ .(. ~ 79 20 o 45 40 20 . Logsin(.-^i)i%.|5-;-- 45 39 55=-PZ (s-PS)= 9.3215800 LpgcosecPZ =10.1447400 cosecPS =10.0075700 2)39'32836oo i9"664i8oo=log sin 27" 19' 4-10 2 _54 58=P Equation of time ='-T-f4" True mean time =3 3A ~^ ^^*^ht,me 3 3J J5 •^^tc^slow ^~ /i" ■( ) J- ly Tintg by Star Altitude. and the hour Inl wlTen T f ^ '•^' ^^^ '^^ ^^t^^. 3om. 17s. west " ""^'^"^ °"* P^°--^ to be ah. To find the watch error— Sidereal. toe of. he i„s,a„,= f-~ Subtract the Sidereal time of ^° ^^ ^' mean noon, corrected for longitude .. 22 o 9 Sidereal interval since mean ~ ~ noon „ •_^ ^ 59 12 An-i the watch was 2m. i6l fast.^^""' ^^' '" "'^^" *'"^^- It should be noted thaf .f+k^ J i- body is north it musV L ^Itctd^"^^^^ °^ '^^^^"'^ polar distance. If south tLr . "" ^^ *° ^^^ its to 90-. ^ '''^ dechnation must be added The following formula i* a T,a« altitude and latitude a e emnl^^'^"'''"'"'^' °"^' ^^ the Plements. employed instead of their com- Sin«~= £?s ^ sin (s-a) 2 c^sTmTps '^"^'"^ « is the altitude, I the latitude. PS the polar distance, and s=:^-±l±l^ TO PIKD THK TIME BV A MHRIOUN TRANSIT OP X HEAVENLY BODY «- on a s.ve.. T^.^ir- ^ -Xre"; ^ 7'iwc hy Meridian Transits. the telescope directed on a distant point or ml^rTnT and the horizontal plate clamped. The telescope will now, if moved in altitude, keep in the plane of the meri- dian, provided the instrument is in adjustment ; and the mstant of transit of any object across the vertical wire or intersection being noted, we can deduce the true time. As the altitude of an object at transit is equal to the alti- tude of the intersection of the meridian and equator plus or minus the declination of the object, we have the equation Altitude = 90° - latitude ± declination, and can set the telescope beforehand at the required altitude. If the latter is more than 50° a diagonal eye piece IS necessary with most instruments. In the case of the sun we may either take the mean of the observed instants of transit of the east and west limbs, or take the transit of one limb and add or subtract the time required for his semi-diameter to pass the meridian (which we obtain from the Almanac). We now have the watch time of transit of the sun's centre, which takes place at appar- ent noon, and have only to find the true mean tim. of ap- parent noon by adding to or subtracting from the latter the equation of time (corrected for longitude), when the ditterence will give us the watch error. Example— At Kingston, on the 2nd of May, 1882 the transit of the sun's west lir^b was observed at iih. 55m A. M. What was the watch error ? Here we have Watch time of transit of limb nh iccm ,^o lime ot the semi-diameter passing the meridian , im. 6s. Watch time of transit of sun's centre iih. 56n7~6r The equation of time, corrected for longitude, was 3m. 11.5s, to be subtracted from apparent time. T^^^**' by Meridian Transits. ,- Apparent time of transit of "^c^ntre r^h. om os"" B.quation of time... om. os. -J jm. 11.5s. True mean time of transit TTi 7 ;; Watch time of transit h' 5^;"-^?-5s. ■__" 5"'"- 6.0s. Watch slow ' " ' .IV^;^"''^^''^'^''''^^'^ i^ the sidereal" fme of the instant of transit, and by workin. out the ^o^spondm, mean time in the usua^l wa.v w! ^e"; Z If a planet is to be observed we take its nVht ascension ^r c^d inThT' ; """''"^ '' '^' '^"^''"^^ '- ^he way airected m the explanations at the end. For observing objects at night the theodolite ought to have an il uminatmg apparatus to light up the v^rel The plan of throwing the light of a la^eru Huougrhe object glass is objectionable if it can avoided It should be noted that the nea. r the observed object s to the .en.th the less will be the effect of any error in he direction of the north and south line, and the greate" that^of one of the telescope p.vots being higher than the With small instrumen's. objects near the equator from poIeTe^hVwr ' ''' '''''-''-' -''' thoselearllt^r if M CHAPTER IV. object a littl. Irf .. telescope directed on the ho^om Wirt rth n"rr T ^"""' ''•''"■ ^"^ lower iimb iTthe sun r T . '°"''' ""^ °''J«'« (""= withita"itrit b;:::„i:rtt"^^ "''" '" ^-'^^^ the vertical arc tl,.7l ^ slow-motion screw of required When the ohrr.'""^ '"°''' '"^'^% »= altitude it wHl re Jtrr^^u''".'""'™'' "« 8^=«est wire, whelThl'^erafrif rTaV'oT '' Vhe" 't1' """ ''' then at once turned over tL , , ' ° telescope is altitude again read. The teTronhlr^^l: '"' '"^ be the apparent altitude ofTheo^ect ''"''"^' ™" theTo" r'"^ ""' ''"""' ^"'i ='«'fl'='al horizon we bnW mot on^rf .m2rc"'"' r " "^'^ '""" ^^ "^ "^-l- is read off ^ '"" '° '"P"«=> "^en the vernier meridlaraUitXtr "'I"' "='» ^Hed. and thetrue the latitudirrii'i',:;;::::"'- "-'"•'••■-''•-ed. Latitude by a Meridian Altitude. ', AND DE OF r sex- hould he in- n the The : (the atact iW of ly as atest bthe 36 is [the will ring ilovv nier true led, ^n Fig. 8 let A be the observer's position, PA q pr , ^ 3^^^j^^ of the earth passing through ^. and the poles (^^1) and therefore in the plane of the meridian. Let O be the earth's centre and let f O 9 be perpendicular to ;i Ai- then . and q will will be the inter- •he meridian. H R will be ,„ f ""V'° '" ""' P'»°' <>' and Win He due non^Z Z^ '^^Tl '"r" observed and let if« H...T,- . , "^ ""^ object parallel ,0 *>. aid A o „ ,7 ""^ "°"''- ^'='»' ^ P direction of the p"! of thT '° u '' ""^ "™ "-^ ">e .he intersection'o the trS '"T ""' ^ « '"« "' O A and produce i .oTthr.enTth 'TT"'"- -f-" angles to H R, and P A Q istlol ,'ht anl " l\T the measured altitude of the obiect and SAO i, ^ , " tion. Now the latif,„(. t a , and b A Q its declma- .hezenith. Also/z A Q=V-P /z p'rii""™ "' .he altitude of the visibg pi It^t^^'"'^^^ '-"'""'==^*2=SO°-0AR=9O»-(SAR-SA0, ,,,. ^. -9'>'--a"itude + declination. hat tt^r.?R^ r a^::r'rd ^?^ i-^r 37 38 Latitude. 2nd case. If the object culminates between the zenith and the visible pole, as at S in figure g. its altitude will be b A H, and we shall have : Latitude=P A H = S A H— S A P = S A H— (go — S A Q) =altitude + declination— go°. If the object is a star which never sets, but de- scribes a diurnal circle round the pole, it will cross the meridian twice in the 24 hours, and we may take its altitude at what is known as its lower transit, as at S*, Fig, 9. Here we have : Latitudes? AH = Si AH + 51 A P=altitude + go°— Figg. declination. Therefore, in the case of such a star we have : Latitude=star's altitude±star's polar distance ; the positive sign being taken if the star is observed below the pole, and vice versa. ""* Case 2 can only apply to the sun when A is within the tropics. In many books on astronomy the formuloe of case I are made to apply to the sun in every siturtion whereas they manifestly fail when he culminates between the zenith and the visible pole. In the Nautical Almanac is given a very simple method ot finding the latitude from an altitude of the pole star taken at any point of its diurnal circle round the pole The time of the observation has to be noted and the cor- responding sidereal time calculated. LONGITUDE. Longitude cannot, like latitude, be measured absolutely, ■'-' ct i^^rr-jt A /'.'^ r . /^^ yCv >fC ^ ,v»^?"-?- - _. i i Longitude. _ as it has no natural zero or origin, and we have^7ss^ an initial meridian arbitrarily, the Enghsh adopting chat of Greenwich But the difference of longitude of two diin'' T ?^' ^' ^°""^- '^^^ ^^-^Pl^^t "method of doing this is by comparing the local time at the two places for the same instant. This is done by signal of some kind or other, such as flashing the sun's rays from station to station, or by the electric telegraph. Since the earth revolves through 360 degrees of hour angle m 34 hours it will pass through 15 degrees in i hour That is when It is one o'clock in the afternoon at a certain ast^on "f f^ ' °'''°^'' '' ^^^*'°" ^5 degrees to the east of It. Ffteenmin,..;. .nd seconds of longitude in arc ^e equivalent to on. ....e and second of time re;:" tively. Tl us applies to sidereal as well as to mean time. That IS If the sidereal clock showed i hour at one place i IS wor h thinking out, for it is often a pu.zle to beginn^ Therefore, if at a preconcerted signal the observers at two stations note the exact local time, either mean or sidereal! tnl rr ""^'^^ '^° ^'" ^'^^ '^^ ^•^^'•-"-e of longi- tude Ordinary watches may be used if their exact rate and their error at any given instant are known. The best way, however, is by telegraphing star transits. If the eastern observer signals at the instant that a certain star IS on the meridian, and the western observer not " ,the time of the signal by his sidereal chronometer and afterwards takes the time of the star's transi at his own station, the interval of time between the t " ra lit allowing for the clock's rate, will evidently give the differ ence o longitude. If the eastern observer notes the fme of he transit, and the western signals the transit at hi station, the same result will be obtained, and by takin' the mean any time lost in the transmission of the signals wUl be corrected; for it is evident that in the first eat the time lost will make the difference of longitude too •he results .vi« be ve-yrerrThr,;;,;'^"- '"^ "-" of den, .ha, for s/4^"^ f„X^/ - '^<''ue.r longitude by find,"; "he shl s'T T '"'°'' """-" analt,.udeofthesunvvhe„tl,!l ^. "' """" '™e by from the meridian, and co^parin 'T'-f ""''^"-^ho"--' keepng Greenwich mean ,Z h ',' "".' <>'>-'o"ome,er he ms,ant the altitude is tal^ I,"' "^'"^ ""^^ -' less ,0 remark tha, when J^^'u ^'^"" ="''"0^' "eed- «'on for r.,e must alwa;: be^^ptd™"'"''"^ "' ^°-«- ^° ™" ™^ ^^=;-- - ^«o„ „s ^ye have to find the angle 1^1 °k '^^ *"^"^J« ? ^ s! threesidesofthetrian^e gtn and "' '^ '^^^°^^' '^^ the formula ^ ^''^^"' ^"^ may therefore use Sin» J? =, ^!lLt:ilSlSin(s~-Z P) A XI. ^ Sin Z S Sin ZP Another formula that may be emp,o;ed'^;3: Azimuth by an Altitude. 41 Cos» — = Cos s Cos (s-s P) Sec X Sec a where a is the al.i.„de of .he object s P its polar dis- tance, ,! the latitude, and s= g +-' + PS asI^iL'ST'" '' T^" "" -o^t convenient of the two as It entails less subtraction than the other. ^J^^Si^tfjr^^^ in.por.ance to sur- ^"■^h as a7ran . theodoite ^Th! f"""' .■"='™™™'. .tX"'th^ f,<-^- ":anit%:rrth-' ^t hori^al^r-- ™ -;tr "'"^- •--- turning it on the heavenly body and taking Its altitude and the horizontal plate reading. It is better to repeat the observation in reversed positions of the instrument and take the mean of the twosetsof readings. The differ- ence of the horizontal readings on the Ime and the heavenly body gives the angle A Z S, and the triangle T' 7 ^ u !^ '°- the anrfp P 7 c u ^^^^^ ^ ^ S when solved gives tne angle i' Z S, whence we have A 7 P th^ ! bearing, and therefore Z P the directtn f I '^"^^''^^ In taking an alt-azimuth of th ul -f 1^^^^^^ altitude only we must add or subtract' th.! / ''"^^" to get the altitude of the cent e If hll ''u''*'' vertical and horizontal wire the snn't °P' ^^' ^ tangential to both. ToTet th. T Z""^^' '' "^'^" semi-diameter we m„.f !. get the lateral correction for latter by thfseP:; tTe^^S^.t 'tZ^' '' ''' tions are got ri(fV observing IhT ^'^ ''°'"^^- rants of the crosVLw S^^^^^^ Tt ^"^'■ edge of the sun tangential to LI I S, '' *° ^^^^ ''"^ of a slow-motion scfrmthHthr.^r '^ "^^"^ tangential. Thus. ZZilht llf V^" ^'''''^'' ^^'^ vertical wire a itX ! '^^ '"" ^^^^^^p the <*' wire a httle and keep it tan^enti-l *^ -^ m 42 Azimuth by an Altitude. Pig- II. « * a little, and using horizontal wire bv the vertiVal cir>,., ^ 4.^ ' ' just touches the former If ^'°"-'"°*-" ^-«- till it the wires of the theodolite are arranged as in Fig. ii we take the observation as follows. Suppose the time to be fore- noon and the apparent motion of the sun in the direction of the arrow. For the first ob- servation get the sun tangen- tial to the wires a b and e / |n the uppper position. This is done by making it overlap ^ „ a jitti^. »n^ • the vertical arc slow-motion "^crew to ke ' 2t T'"^ edge tangential to the horizontal wirL /uShI 7" touches a b, when the verniers are read^T TheTnst " tr'irfsTrr ^-^ ^'^^-'^ -^'^ ^^^o^ w:::rp;:t;ti°::-r:rt:te^xrf gential to a b. The mean of the two a t^es fs' a^^^^^^^ and also that of the two horizontal readings The W mus also be noted so as to correct the decLation The r^rs^st^^nr ^'^ ' '-'- ^°^^ ^^^-' -^ Ma^rci ^rRnl^'Tl '" ^'*'*"^' ^^' ''' 4°" ^n the 3rd of March 1882, at sh. 30m. p.m., or yh. 36m. Greenwich mean t,me, two altitudes were taken of the sun wTh a transit theodolite in reversed positions for the purpo e of testmg the accuracy of a north and south line the horf zontal arc being first clamped at zero, and the telesco"" directed northwards along the line. leiescope READINGS ON SUN. Tof r.u X- Altitude. Azimnth. 1st observatlon-310 8' 220- o' Mean ^° '^ " ^^^^ ^^ Mean 30 42 330 g Azimuth by an Altitude. 43 To Correct the Altitude t« /^ . .^ ^ Aiuuae To Correct the Declination. Pofro .• 30 42 o Decimation at Mean) ^o ParSax+~ ' f .^°°" ^* Greenwich ^ 44' 37" S raralJax-l-- 8 Correction for 7A hrs ) Truealtitudeao 40 32 at-57". 5 per^hou;} 7 3i S Formula used ; True declination 6 37 6 S 90 Sun's N. P. D.=96 37 6 Cos.»~=cos. s. cos. (s— S P) sec. ; sec. a "= 30* 40' 32' ;= 44 13 40 S P= 96 27 6 2)171 31 18 *= S5 45 39 96 37 6 s— S P=^o 51 27 log cos s= 8.8687314 ogcos(s-SP)= 9.9921540 ogseca =10.0654637 log sec X =10.1447380 2)39.0710881 , 19-5355440 ==io+logcos69''56' 2 -2=139* 52 360 o Sun's Azimuth = 220* 3' The direction of the line was therefore ^. " The formula — Cos ,PZS =cos s cos (s-PS) sec ; sec a where PS is the sun's (or star's) polar distance, a its alti- .PS-h/l+a . tude, k the latitude, and s: , is thus derived. We have, in the triangle PZS, if ^'- P^+PS+ZS o. Cos»?^= si" ^' sin (s'-PS) 2 sm PZsinZS Sin (s'_PS)=cos {90-(/-^-PS)f If A m I If 44 Meridian by Equal Altitudes of a Star. =cos j8o~PZ-ZS + PS =cos-22±P^±9o^:ZS + PS =cos ^+a+PS =cos s COS (s- PS)= cos2£l^^^±9£:i:^S--PS = cos|qo_?^±PS+ZS| "" . , Therefore cos»^£?~ ^^^ s cos (s— PS) c:^M 1 . ^ cos ^ cos o Similarly it may be shown that sin»^?^= cosjjin (s—a) 2 cos ^ sin PS scope on it abo.ff 7 t ^^^odohte direct the tele- height. i^;\'2l:T.itr ' ^^^ ^-*^^* tion screws cet thpc/ , ^ ""^^"^ °^^^^ ^'o^ mo- wires. HaWnTtakeTtVe"^^^^^^ ''; i"*^"^^*'°" °^ ^^^ leave the vertica one ',/ '- '"^ ^°"'°"*"^ """' tal plate and Took . T^' u' '°°''" *^^ "PP^^ ^^"^on- i=:;;.ls:sr,£H ■F■■"•-■ the plate at thTf ^ a- '^''"''*'°" °^*^^ '"e^'dian. Set lantet" cJ^tSttT.^^^^^^^^^ tT ^?^^"^^" ^^^^^ wires, and drive ir p cLts at fhf 1 . '"^^'-^^^^^^ ^^ ^^e station. ^ * *^^ ^^"*^™ ^"^ theodoHte This method is rai-hpr o 4..j-'/<'^«'"^''^<^i"^<^<^y '«*»'^'V7,i'-w shortened by observing th-^^^^^ °"'' *^ ^' "^'^ ^^ y oDservmg the star when nearer the meridian. i 1 % 1 Meridim 5> Transit ofPoh Star. TO PXKD ^^-^«i;iAN b/,;;^^^^ ■eetaSTAR AT ITS MERIDIAN TRANSIT ^ ^°^^ star will be o„ te terfd L„ " '"''' k '' ""'"' "■' P°'' and take .he reading „f the ho zo„ a p le Xv" " the telescope and horizontal plate and dir,.! it f"^ pn the star at the same intervaUf'time after hf '"^^ in this instance 4 minutes after th. «! 1 "'"sil- again read the plate tL "'^ /f "bservation-and gfve the true north "" °' "" '"° «''«°«^ »«' o/t^LinXrvtsrorhTte^r '" --"-^^^ poSa^ ^rve^ r; .r "l- ^'-^ *"-"• ""^ '-e niotion appear 7 Z^t^^i^ZZl tZ ^ when at their fn-PafAe+ ^00* ""'^ time two read nes takf^n if ,.,« u mean ot the horizontal pTatl reading when tl: tX'""'' '^''" '"^ on some well-deflned dltl« obiect t "'f ™' '""'" we can now obtain the ^■,rnlZl:L:j;::ZLT 'k 46 Meridian by Greatest Elongation. In Fig. 12 Jet the plane of the paper let Z be the observer's position. A the refernngmarl^.P the pole, and S the star at Its greatest eastern elongation. P z S aTl':nVee^SVK!^-^^^^-^'^^ SinPZS=^ilLP__§ ■ ^r-^.^^ SinPZS=^°lJ Uiu /.■ v^ince P S is the complementing A . A. ^o,. ^^i! ^ the complement of the ti. i^'% "l'"^.*'-' -^ P*g' 13. ' ^ pot bever^ accrately known ^^^ '^''""•^ ""d A- /J^. ^r-^ic -^ "*e complement of the h //..?^..^°t bevery accurately known. -- r^^ /^e .X^.>^°w^ having from this equation nrevin., i r ^ .^._ ^>«^ni:Ie P Z S. and having obtained Az^r^ T^ *^^ ^^ .^.r-dings. we get at once^he aTg^e a Z P wh^'n ''^^'^^^ -^^^ ^^ ^ ''""■ed azimuth. ^ ^ ^ ^' "^h'ch IS the re. Tf 1 solvmg the equation. greatest elongation by , I,- u . Cos Z P S=cot «J tan ; which gives us the star's hour angle "d h of the observation. ^ ' ""^ ^^"^^ ^he time The altitude is given by the equation. Sin. altitude=~!iili T/- ., . . sin 8 " It IS inconvenient to observp th^ \ greatest elongation we can use the f M " "'^l '' ''' which is approximately tru. in thect f °^'"^ ^°™"J^ the pole. ^ " ^''^ ^^^^ o^ a star very near tan A* , , t~^^X-=''" ^ P S where A^ is the star's azimuth Z P q v u at the time of observation Th a\ '*' ^^^'^ ^"^'^ elongation. '"'^^t'°"' ^^^ A.its azimuth at greatest Meridian by High and Low Stars. Choose two stars differing but little in nVhf o one of which culminates near the enith fnH ^"'7' near the south horizon (or the north hn ^ southern hemisphere.) Level the th ^ T'°" '^ '" *^" fully. The ereat rimt ? theodolite very care- of the telescoTe wi , 1'" h' "' J ^'^ ^""^'"^t'^" ^-e -nith, howeve'faT m. K . "'*' ^'^ "^"^'^" ^^ ^^e "PPer star, noting the watch time Th! ' n ■"'° watch error approximatelv .nH J!^ "''" «""= ">« approximate wa.cht^mt at whrh t ' ,""" """^ '"^ transit. By keeninsr fh. ,.l '"""■ *'" *'" .ha. instant arXef we shlr/t >""' °" ">" ^'"«" plane of the meridian r^n by ^pe^ inTfhe":"'^ '" '"t another pair of hiVh ^n^ i ^/^P®^""& the process with «o.onH^m:r?;if„r;:::r:„ts:"''"''''^^'^^^^ wit^^rdiior^eye^nr'^ThV'^""' r""""' ««'« are to thezeni.hThe &. "" '"' "PP" «"» The Canadian Government Manual of <;.,. '"ends for azimuth the formula : '^ '''°"^- tan. P Z S==^-^^LL^J^SJl^B.^JPS i-tanPStan>lcosZPS as applied to observations of the Dole st^r- u . u qmres special tables in oraer to work'uou^^^^^ "* '* "" ri :f \w Then CHAPTER V. Sl/y^ DIALS. To a person acquainted with the ri,rl,-m^„* r * be th. sun's decUwLn Th"T f "^'^ ""' *'■^'"- Poini in the line will ,^' t " °' ""J" P^'icular will always IieT„.r'sars.::it t^r'"" ™™^' ""' angle. On .l,is principal a|f„ i? "^ ^"'" ''°" The position of the shadow 1, 1 ^'^ constructed. at the instant and VT T ^^ ""^ '""'^ ^our angle time; so'harin oil f r '"^''T "'^ "*"'■'"' -^" have to app,; l^^jLtAl:.'"'""^ -"" '™' ^ Dials are geneiaily either horizontal or vertical Tn he former case the shadow of the stile Ti, T n j thrown on a horizontal „i . f ' ' '^ called, is ^^^^ honzontal plate ; m the latter on a vertical woll'd^trtoyti™:.:' ''°"^™'^' "'^'^ ^™^"'°- which shadow lines would be paVirfor .^^ '°"'^' "^^^ tances apart for eau!l .n. , / '*'^^' ^"'^ *^^>^ ^''^■ part lor equal intervals of time would rapidlv %'\ 50 Sun Dials. li n increase according to the sun's distance from the meridian and wouM become indefinitely great when he was on th ' At the poles the stile would be a fine vertical rod from the base of which 24 straight lines, radiating at inter^k 1 I 'tt^'T- """^ '"'i'^^'' ">« hours. The hne „„ ."o Gre !' '1"'°" "" "'-■°™ « "-^ '™^ corresponding fhof V ; , «*feei--?faees the stile must be set so hat us angle of elevation above the horizontal plane s the same as the latitude of the place. ' HORIZONTAL DIALS. metalstile^d on a horizontal plate on the top of a pillar. Fig. 13 is an elevation and Fig. 14 a plan. The angle of elevation of the stile is made equal to the latitude of the place, and if the variation of the compass is known, the latter may be used to get the dial with its stile in the plane of the meridian. The hour lines on the plate are marked out thus : let A B (Fig 14) be the base of the stile, P'g- 13- Pfg' 14- 'i» ■ meridian, vas on the rod, from intervals e line on isponding 5 the 2ero be set so plane is riangular op of a ind Fig. ation of latitude n of the may be e in the ur lines lus : let he stile, Sun Dials. line ERF n ^ f Through B draw a straight line iL BF perpendicular to A B D Frnm n a i Da, Dal D A n Ai ;? ^ o u. l^rom Ddrawhnes «, u a , u b, D b\ &c., meetmg E B F in a nt Xr.. and making the angles B D «. B D a^ a D /' /'n m" &c., each equal to 15 degrees From' a J ' * ' turned up on c c^ till it abuts on B C when Dw II cde with C, and A C will be parallel 'tnVT , • '''''"' perpendicular to the plane of D."'. ^^^P^^^^^^^s and When the divisions on the line E B F mn ««• *v, , we continue them thus • In A c wL , ^ ^^^^^ any point .,and throughit draw\% l';^!^; JT^^ ]t 9 A. M. line) meeting A b^, A a^ &c fn T i ^. ' ^ ! make oy equal to 1 ^1 to o\^^' t' t' *^" ^"^ &c., draw straight lines A . /!'. t' ""^. *^°"S:h y, ^S evening hour lines. '' ^ ' *"' ^^^^^ ^"^ ^e the onlhfoltTdr'^" "^^^^ ^°"^^ -rnmghour lines VERTICAL DIALS. a plan as any s to fix a flat H,-<=L- h. • P'® 52 Vertical Sun Dials. P'g' 15- The disk should be roughly per- pendicular to the sun's rays at noon about the equinoxes. -&e The bright spot in the middle of the shadow of the disk on the wall indicates the hour. The hour lines are found thus : At the time the sun is on the meri- dian mark the position of the bright spot on the wall. Let A be the hole in the disk and B the spot. Measure A B. Through distance minus tlie cclatol M . .u "™ ' P°'" would coindde and C D r Ji '^/ ""'"'^ ^ »« ^ "ne C A, „.u1d' bTa^,," [^'pr r^f I??? a watch, set to noon at the tims nfVh! '^''* -ark the positions of thlspo. on. hf :a,""a: h""^"' ^"^ ivehours. Straio-ht li„„ • .., ^ "^^" at the success, the hour lines '"nes jo.mng these points with C will be Of course a large triangular stile C A R ™- i.. .. substituted for the disk ; or'we J^ use a r!d cf « ^ J ^ %i. CHAPTER VI. MEMAnKS O^ PORTABLE A^TnO^MlCAL I^TnUME^TS. THE REFRACTING TELESCOPE other. The W ImT '"^ "" ''' ^^''^ ^* '^- object at its focuT n th. '' '" "^'"^'^ ^'"^^^ °^ ^^e J ai itb locus m the same way a<5 the Ipnc r.f ^ u tographic camera, and the eye piece simt^ S ^T magn fy this imajrp tu^ • ^ ^impy serves to natural Position a UiXol": if^' '" "' by means of which the inv.rfo^ ■ employed, This has. however thV H . '"''^" '' "^"^" '"^^^t«<^- used in telescopes desi/n dfor Ih^ " '"' ^^""^"'^ objects without LkTnf ""^"^ examination of ''positive 'n \ I u^ measurements. Secondly, the positive, ,n which the common focus of thp f w^ i IS outside the eye piece Thi. il T Vf '^"'^^ telescopes intended "^ormea^unL It u ""' '" ^" spider lines &c wi ^^^f""^'"^ ^"S:les by means of focused he image an?sTstel;"^'"""^*'' ^^^P^^^ nage and system of wires are in the same 54 The Telescope. piece 'VJ!l'n°T°" h"' °^ '^' °^J^^* ^^^'^ -nd eye ' piece. The position of the focus of the former fUr..S on the distance of the object-that of th^t L f ht shnA Jr ? ^^' '"^ ^' ^^^J"^*^^ t° s"it the observer -short-sighted people having to push the eye piece in wh.Ie t ose who have long sight require a longer focus The larger the object glass is the more rays from the object are collected on the image, and the bngh eTit t The greater the magnifying power of the eye piece t^e more apparent are any defects of definition in th^e l^gt The magnifying power of the telescope is measured by the fraction f-S5L]55gth_of^^ectglass. , ^ focal length of eye piece ^""^' '^ this fraction were 4, the linear dimensions (if the object seen piece at the centre of tne object glass. In large telescopes the field of view is so small that it s necessar;- to use a "finder," which is simpTy a sma be prrXr""'' *" " ^° "■^' "•» "- °f •"' 'woXll A diagonal eye piece is one in which there is a mirror or pnsm between its two lenses by which the rays o7feh oflr/nVoffhl ^"''" ^"^ ''""'^ '-- «■' ^'^^^ 01 tne end of the eye piece. It is used for observing objects when the altitude is so great that it wouTd bf sTopetbt"'' " '"""'''''" '" '^^ "P 'O™^- «>" tie! Lenses have to be corrected for chromatic aberration and ^Pl,cr.cal aberration. Take the case of an obje ^S co„ and eye ■ depends ■r on the 'ery indi- observer piece in, focus, rom the ter it is. •iece the i inrage. ured by if this ict seen It they erefore, IS the ,^/Tu^sX/v i is, in the eye that it small shall Tor or ■ light istead ;rving Id be 1 tele- t and con- The Telescope. — — ■■■ 00 'isting of a simple glass lens. Ra^^TTiiriT^Tdiflfe^ appearance of coloured light Thi chmZ °',"^"""'.°* is go. rid of by using a cf^pound et crLt,tT,'°" other. Such a lens is called "achromatic." By "spherical aberration" is meant the disnersion „f rays caused by the central portion of a lens wZ Xrica su aces havmga different focus from its outeror mar^ Sn combil-r "^^ "^ '"^ '"- ^-" '° '"^ '■'- The way to ascertain if the obiect riass hp. t,« properly corrected for colour is to tumt on some brigh" object, such as the moon or Jupiter and out th, rr irp-;L?rofd?ar„r '"-• — ^^^^^^^ portf^t-::rj:h^:'ru£ focusmg ,t on an object, afterwards removingThe d^sk r=„u^;-r.htstr---p-'»^^^ irradiation, or ™^, ^t one side "'^'" """■ ^" THE MICROMETER. anjutrtZcr i.^rrrrrof ---r--" -y theodolite, placed tl tt:ZlTfZZu,:T': glass and eye piece of a telescope, or o,. hi "read' microscope " which will be descrbed presently Thf pnncple of the micrometer is simply thisf Snp^o'se ,1a 56 The Micrometer. a point-such as the intersection of the cross wires-can LetTbrthr",""=H''" " "^* "^ "-- °f "-W the wres through th,s space ; it is evident that on. lam of the scr«w will move them through an angle -?-. lark"" tt °' "" '"■'" """■''' °' """' "^''^ ^" i"d« to niark the commencement and ead of each turn. If the head ,s made large er,o-.«h to enable its rim to be divMed ■nto „ equa^ parts we sl..a h,v,e the means of muring °" "«'° W^ ^y '""■■"'"S '■ « head through one division. Thus by making the thread of the screw sufficientlv fine aj,d ,ts head large enough, we have the means oTmeasur-' ided we ."' t, '° "" ^"'^'""^ "^^'^^ of accuracy, pro- This I„ r ""■ ""?"'" ''"'"'= °f °"^'"" oftheLL. Tms may be ascertan>ed by finding how many turns i take., to move the wire across the image of an obiect of know.., dimensions at a known distance A leveufng rod amded by the distance, gives, of course, the chord of the v.et TT^- ^"r '' "^-^"^ ^ ^^="^ '" *'>^ fi °W of thesWew. ™' "''"^'' '=°"^^'"'"^ '° 'h-'-nsof /^/^. l6. There are several different forms of micrometer. The accompanymg figure (i6) represeru... ,^he one known as the filar (or thread) micrometer. T . parallel wires mm It' k w . s wires— can 3 of a screw. i let n be the "ed to move fiat one turn n anele — n an index to urn. If the o be divided f measuring Jne division. ciently fine, 5 of measur- :uracy, pro- f the screw. my turns it m object of veiling rod loved over, fiord of the he field of he turns of ter. The wn as the ires m m, ^ a ^^^ Micrometer. « n, .a« fixed to two frampc /» aTT^ ''iljer, vvhich are tZVaT ' ' '^'"^'"^ °"^ ^^^^in the ".e wire ,„ «. and is moved by The serew A .T I"?", "-dve to measure the anralar Hie, T Suppose we .ontal wire. If „ow the sc ^ A i t'eTtm ^^ '"'"■ -e'earoist'"'' ^f.'"^ °^"^e dllret! and ^'^::^^:^:^^i^^:^^:i '"-= °' ^ lo „ „. This is not theexact Z^hl c ""'"^^ '" *» "P »"' i. serves .o illus.rlrhe prLcip,:'''""'"^^ f°"°-''' THE READING MICROSCOPE a »tret/i:r;ivv„d''^ '' r^" '^'^'-"^ -•" for reading the frac^ a, pa J oT hTd"' °' "^"'" graduated circles of l»r J ■, divisions of the is fixed, the c^^: beii?:. a"ete7t:"l t^^ "'^"^^""^ ■"strnment and moving withTtVh '°'" "^ ""' one screw and moveable ramt Jj't "'"""^'^^ has only ofcross-wires in the comm™% /''""'""S ^ pair and eye piece of the micrrcop ""Th "' '"'^^^' ^'^^^ "sed in exactly the ^Z ^ T "'^ cross-wires are telescope, only tha. the oh"''', "• """'^ of a theodolite arc, o„ which fhm CO co'r„ Tf '= "'^ ^-^uated To n, .e the matter Ce^we I" ' tlTT' "^ '°^'''=''- measurement of a hori2o-,f»l = "' t. "''' "== of the the arc of „hioh is gX .Id f '%„' ^- '"'' "'^°<'°'''» that one turn of the n^cromete, screw 1^^"'"-, ^"^^ °^^ ■nmute, and that its head iT7- IT- ""Sf'^alent to one 111 58 The Reading Microscope. with one of the objects. On viewing the arc through the microscope (which it must be remembered is a fixture) the wire intersection of the latter must be made to coin- cide with one of the divisions of the arc by means of the micrometer screw, and the reading of the index of the latter noted. Suppose the arc reading to be io° 20', and that of the screw head 15". Now move the circle and bring the telescope to bear on the other object. The cross-wires of the microscope will probably fall some- where between two divisions of the arc, say between 50* 30 and 50- 40'. Turn the screw till the cross-wire is on the 50 30 division, and suppose that it takes between three and lour turns, and that the index marks 25'. The micrometer wire will have been moved 3' 10", and the true reading of the second object will be 50' ^^ 10". The angle measured is therefore 40° 13' 10". Two or more reading microscopes are placed at equal distances round the circle, and the readings of all taken Errors due to eccentricity are thus got rid of, and those due to faulty graduation and observation much diminished. THE SPIRIT LEVEL. The spirit level is used, not only to bring certain lines of an instrument as nearly as possible into a horizontal position, but also to measure the deviation of these lines from the horizontal. For this purpose the glass tube is graduated, usually from its middle towards both ends and the reading of the ends of the air bubble noted. The length of the bubble depends upon the temperature, and the latter should therefore be also noted. To obtain the value of one division of the level— th?it is of the vertical angle through which the level must be moved in order that the ends of the bubble may be dis- placed one division-a simple plan is to rest the level upon some support (such as the horizontal plate of a theodolite) that can be moved vertically and which is con- T^he spirit Level. —- — 59 nected with a telp<5ron« tu i " scope directed on a^e«iJf' "'" " ''"="="'> '"' •«!'- known distance Lh T "'^""""g '"d set up at a bubble tTo"he tee::etir om:;'.'''' r' "' "« the roH Th^ u 1 "''^^^^"°" oi the homonta wire with by n::ans omet^ot sZ^flSr ' ''/" .T'' ""'-"' moved a certain numbrrTf h ' ' °'"'' '""''''' h"' in. of the telescoprwt oltfrod ~"' '^ ''''' "^O" difference of the rod rLn ? '^ "°" "°'=<'- The gives the chord of "he verti T T''' '^ "= <''^'='"«' and this divided .; t ;i::: t "^tTo^^^j^tor ' Ro;"al'Mim::;^'o';ri:t:lf; --^ -.escope at the 383 feet a ver'icalX l^: 'Z ' ='! '' ^ t'''"" "' 20 divisions altered the r?»T "''f ""d the bubble feet. The resul .nfdecim^ "" ' "''"'"" '''^ "''^ tangent of the subtfndedTn'leTr °;T^'^ '' '"' made th. value of the latter 6?.45 ""'°- "''''''' on wh- irvttteTrll'rh'' -'^"^ ^f"' " ">--'- longer than the otL he bubtr"",'"'' "?"' °"= '=« '^ is highest will be greates and, T fh 7 f ■"■' "" "'"^'' for end the bubble readTntwHIK "' " '""'<' ^"^ other hand, if the lei te L 'w"»'" P'""=- O" «h« on which they rest rnUetr" "the blw '"' 'H' '""'" remain the same on reversal VL, v ■'"''"'«' ""'" of the level A and the other B I "' , "" "" ™^ ^-"^ the reading of A the hrgesti^\''',V'™' ."'<'■■ "ill make adjusted ifvel rest! o^a fonf^S'T"^-.. '^ ^ f"'^ a^t^^iLrerS'-^-"--^^^^^^^^ esptiri;trhe":afro7t "'l^trT? ^°'"^'-<'' easily get out of adjustment Th; ° i""='' "''''='• -^ace tested by tie ^3 ^bt^TtCf ^ tlftt 6o The Spirit In case of the pivotsof a transa tele: .ope. Placing the level upon them take the readinj^-. of the bubble ends, and call the reading next the west pivot W and the other E Then reversing the level lake the readings over again and call them W^ and E'. The number of divisions by v.nich the bubble is displaced by the difference ol level of the pivots is given by the formula : 4 To find the achud slope of the pivots we must multiply this quantity by the value of a division of the level. The level error i obtained separately by simply chang- ing the signs of W and E^ in the above formula, when we have : W— Wi— E + E' W-E iWj— El) 4 Level error = Of course If W+Wi^E + Ei there is no slope, and. in practice, when (he level is out of adjustment, we may get the points of support horizontal by raising one of them till this IS the case: For instance; if W were 20 and E 10, W^ would have to be 10 and Ei 20. If the level is in adjustment we must have W— E=W' -El In this case we have only to .ake W and E and the slope IS obtained from the formula W-E ~ — X value of one division Example-Take W=25, E=io, Wi = i5, £> 20 Value of one division =6". Here we have^^^ti_5i::i2r7lO-_ ip Multiplying this by 6 ' we havt 15" as the Lpe, the west pivot being f Ki highest. The level error is-^^^=i§=I2±?o x 6"=3o" ^. ^^^ Chronometer. - I^preat accuracy is required ~thIZ~77u ^77 number of times in each do.Th it ^^"^'^ ^^ ""^^^ ^ reading and taking the sa'me "'h ^"'/' "^ ^'^^"^ '^^'^ two positions. The dLrenre fT. °^ ''"'^'"^^ '" ^^e in.s at .He two ends diW Id ; '/.IT ^'T'^^^'' readings will give the slope """'^''" °^ *^« -^^^i^:e:;tt::::.;^^^ wlX'hrrrstXwLottlf ^^^1 .^^^ *^^ -^^- on by means of the adS 'P^''""'^ ^^°^«- Then, both its ends read the same """' "'^^ ^'^ ^"^^^^ *i" Th ^"^ CHRONOMETER. changes of ,e.- ..ratu^tve m,J? ?° '=°"^'^""e"er every "P at the rl,Z l^Z^l '?','.? ''"''" '' -""^ --parte, the sprmg^c^m:: to';;;;;:- ■-;„? 62 The Electro Chronograph. larity of rate may result. If a chronometer has run down It requires a quick rotatory mover- ent to start it after it has been wound. Transporting— On board ship chronometers are allowed to swing freely in their gimbals so that they may keep a horizontal position ; but on land they should be' fastened with a clamp. Pockt ; chronometers should always be kept in the same position, and if carried in the pocket in the day should be hung up at night. Chronometers have usually a different rate when travelling from what they keep when stationary.' The travelling rate may be found by comparing observations for time taken at the same place before and after a journey, or from observations at two places of which the difference of longitude is known. For mean time observations an ordinary watch may be used by comparing it with the chronometer, provided the rate of the watch is known. Chronometers are generally made to beat half seconds. THE ELECTRO CHRONOGRAPH. Under this head may be included all contrivances for registering small intervals of time by visible marks pro- duced by an electro magnet, and thus recording to a precise fraction of a second the actual instant of an occurrance. By this means an observer at a station A can record at a distant station B the exact instant at which a given star passes his meridian, and thus the difference of longitude of the two stations may be ascer- tained. REFLECTINO INSTRUMENTS. THE SEXTANT. A person accustomed to work with the pocket sextant will have little difficulty in using the larger kind ; and the latter, with its adjustments, is so fully described in most The Sextant. 63 0^,!°; .:"™>';"'! "»' 'i"le need bo said aboiTiT^ or merely wi,h „ «r:: i:' o'':: t :':";:':!: '-"f or wi.h a p,a,e of «,ass «oa.i„ron h Z ' Thl: failing eo ca,ch^r;:'i a 'To, ,7:r T^'T" '^ errors caused by wan. of pa^ralieii. r;,/:,,":™'! roof, when one half of a set of observations ha, been ,ake„ the roof shonid be reversed end for e„,f. For akinl .he eth":.h-er Jti'e^h'Tr:- fetU: "Z .l" -"'- noted when .he circles jns. to„ct"X:''.hrre';i,r.t: he images shonid be receding fron, each o.her he Iw tude of .he lower lin,b „,us. be taken in .he fo,^„o' „ and of .he upper Imib in the afternoon For a l„n„T? of .he sun direct the telescope on the n^oon and " or more of the hinged dark glasstrrhrsl""Th: ■ndex error shonM be obtained, and applied asrist": A common fault of the sextan, is that the optical power ab le o r ad°.r " 'T ^'"^"- ^^'^ ^ '""^ "- in belg no. bi^^of .L*;t:'cro;rh'° ^"""^ 'V"^ =>■= "- c ui ine contact of the images within 30". THE SIMPLE REFLECTING CIRCLE. This is simply a sextant with its arc graduated for th. wholearcumferenceofacircle.andwithfhein xaX^^^^ at" : h^nT^The °''"'; T'"^ ^"^ carrv.:ngarni ^ at each end. The mean of the two verniers can be taken ; t Hd ot "xhlr" ^"' "^ ^"°^ '"^ *° eccentncitV : got nd of. This arrangement also tends to diminish the errors of graduation and observation ^^Some reflecting circles have three verniers at intervals Itf 64 The Repeating Reflecting Circle. Ill . 1^ I - *■ THE REPEATING REFLECTING CIRCLE. In the repeating reflecting circle the horizon glass (w Fig. 17), instead of being immovable, is attached to an I arm which revolves about the centre of the instrument and which also carries the telescope (0 and a vernier (v). , The index glass (^f) is , . ^'^- '7- carried on another revolving arm, which also has a vernier vK The arc is graduated from 0° to 720° in the direction of the hands of a clock. To use the instrument the index arm is clamped and Its reading taken. The telescope is then directed on the right hand object (b), the circle revolved till the images coincide, and the telescope arm clamped. The index arm is then undamped, the telescope directed on the left hand object (a), and the index moved forward till the images again coincide, when its vernier is read The difference between the two readings of the index vernier IS twice the angle between the objects. This repeating process may be carried on for any even number of times The hrst and last readings only are taken, and their difference, divided by the number of ^g^^&^ gives the angle. If the angle is changing, as in the case of an alti- tude, the result will be the mean of the angles observed and the time of each observation having been noted the mean of the times is taken. This instrument will not measure a greater angle than the sextant. Its advantages over the latter are that there IS no index error, and errors of reading, graduation, and eccentricity are all nearly r: minated by taking a sufficient number of cross-observations. J^^^'^nmati^ Reflecting Circle. The dip-measurer is a reDpat,V^r~^~^ 77 ' '"irror («) n^ounted on the 1,^"^ "''"'^^ ^^'^^ ^^^ a third height of its silvered portion R ^'^ """'^^^^ ^^e cation an,,es of even roe'han^,,"!^- ^['^- -odifi- and the instrument can thelf \ ""^^ ^^ measured, altitudes of objects near thetnith' "^^ ^- taking double' In the prlLatirr' fl ' ^^-^^HCTmo crKCLE. «-d. andlstead /tteon"'-^ ^'^ J^^^^^^^^ ^^) - ^^^^^^^i^on nnrror there is a small ff^ prismatic reflector (A) [vvhichhalfcovers the object ;f'a^s- The index mirror ('«) ;s carried on an arm which revolves round the centreofthe circle and has a vernier (. ^i) ^^ j^^^j^ ends. This instrument will measure ang^les of any di- [^"ension, and has also the 'Ollowmrr advantages- (i) Eccentricity is completely Fi, ,. J^I'rn.nated by using both images are brighter than inT"''" ^'^ '^^^ ^^^^^^^^ ralyone vernier anH i„T ''"■'=''' '" h^'ving circumference Tm^elLrtr "°' "'''"''"« "'^ *'«"- '>';V-;sno.e„™„rSere"Sr^'^^"''^^'"'=- '"-, .* C.a„ Jnel^rto"',*"^ ™"'°'' "^ "--^ and to use the mean of all the wires instead ofthe tr^nS across the central one. In the field the best plan s for an assistant to hold the watch or chronometer, the ob- erver calhng out "stop" as the object passes e^ch wire In the case of the sun, moon, and planets the insTlnt' noted ,s when the edge of the object touches the w," e. her m commg up to it or leaving it, the time required for Its semi-diameter to pass the meridian being after wards added or s»bt«ete4; ut^Us, Urk ku. o.. .^."..!"> The first adjustment to be attended to is that of colH- mation. This may be effected by getting the central wire on some well-defined distant object, or on a circumpo Ir star at its greatest elongation. The telescope is hen7e versed in its supports, end for end. when, if the wire stni bisects the ob ect. the collimation is all nght. ^^^1)3 " one side of it it must be moved towards it half il ! valbythe collimation screws. Thet^Lm nt 3't en moved laterally by means of small screws conne t d wilh one of the Y supports till the wire bisect. "^^'^'! !^'*^ "'r.* n/t^^^P^ ^s again reversed an he ^ ocet'Te' peated till the collimation is perfect. The horizontality of the axis of the pivots is nKf • a by the striding level and foot screws. '!' he je,' his generally an error of its own which is itself mIh . change (owing to alterations in temptl u^ 2 d ttal flexure &c.) u will be found convenient o eve 4^ pivots by getting them in such a position that he LvJ wUl have equa but opposite readings in reversed ^o" tions. Thus, if m -ne position the east end of th. K VT, reads 10. while the ••. e.. end reads x., th n onlvts' th! east end should re^d 12 and the west 10. ' ^ I I !t t -m t% 1.1 ^ „ '11 ml 1 ! The Transit Telescope. If one of the pivots has a larger diametei* than the oiher it is evident that when their upper surface is level their axis will not be so. This will entail a consiant error which will be investigated presently. The verticality of the central wire must be tested by levelling the pivots and noticing whether the wire re- mains upon the same point throughout its whole length when the telescope is slowly moved in altitude. If the collimation is out of adjustment, but the level- ling correct, the line of cc' nation will sweep but a cone. If the collimation is correct but the levelling in- accurate, it will describe a great circle, but not a ver- tical one. If both are right it will move in a vertical plane. We have now to make this plane coincide with some given one-say that of the meridian. The north and south line may have been already approximately ob- tained by means of a theodolite, and we can now find it exactly by one of the following methods. (i) By transits oi two stars differing little in right as- cension, one as near the pole, the other as far from it as possible. Let a be the rigiit ascension, d the declination and t the observe clock time of transit of the star near the pole; ai, 3^, and t^ the same quantities for the other star, d the azimuth oi the instrument — in other words, the error or deviation to be determined—and f the lati- tude. Then d is found from the formula, ( J cos ^sin ((J— en , ate the effects of vibration prod 'd bv .h° "'" °'''.'- movements the ends of the legs mav h» ^ "•'^^"'er's notches in flat blocks of wood pTaL ' fh .'" '''' '" on a st„.,e setves for the da, titl-l^-^ltUf fb ~ ■t i<\ IMAGE EVALUATION TEST TARGET (MT-3) Us .A. 1.0 I.I |5n ^' £f Ufi 12.0 2.5 2.2 IL25 i 1.4 1.6 Hiotographic Sciences Corporation 23 WEST MAIN STREET WEBSTER, N.Y. 14580 (716) 872-4503 4»^ \ :\ ^ % °- of .he •he .en.,h distance of the equato/r^rSve: ^^ ""^ " and adding, aSTTlfJZp Therefore, if ^ and (^ \, the la,„ude from the diffZ^ZZ^'T.^^ -^^ ''" «nd ■ng their actual values. Moreo J / ' ''"'""" ''"ow- the same the refractions II „° T' -" '"'' "■' "^ nearly and we shall only have toll „,? "'"""''" -^^ «i>er' of the ..fractions at the t,™ :,t"rdes""""' "" ''""^"^^ -ves,ha.„,.ee„hrstcla.:ratTZ:-^ ::;::] 1« f t I f I 1 1 I |i ! 78 The Zenith Telescope. elevation. The latitude must be known approximately, and a pair of sta.s selected which are of so nearly the same meridian zenith distance at that latitude that they will both pass withm the field of view of the telescope without our having to alter its angle of elevation. A^ a rule, z and z must not differ by more than 50' at the most. If the axis is truly vertical and the telescope remains at the same vertical angle at the observation of both stars, then it is plain that the difference of z and z may be read by a micrometer in the eye piece. It is usual to observe only stars which' pass within 25 degrees of the zenith. The telescope has a long diagonal eye piece with a micrometer in its focus, and the micro- meter wire is at right angles to the meridian. There is a very delicate level attached to the telescope, and a vertical arc wh'.ch serves as a finder. By reading th's level at each observation we can detect and allow for any change in the angle of elevation of the telescope. The above is the merest outline of the principle of the instrument, and reference must be mp'^ to other works for the details of its construction 'he method of using it is this: The latitude being aheady approximately known, a pair of stars is found from a star catalogue, both of which will pass within the field of view without altering the elevation, and which have nearly the same right ascension. The reason for this is that their transit may take place within so short in interval of time that the state of the instrument may remain unchanged : but a sufficient interval must be allowed for reading the micrometer and level and reversing in azimuth ; say, not less than one minute or more than twenty. The meridian line must have been previously ascertained by transits of known stars, or otherwise, and the chronometer time calculated at which each of the stars will culminate. The telescope having been brought into the meridian, ready for the star which culminates first, and set for the mean icope. nown approximately, are of so nearly the at latitude that they new of the telescope i of elevation. Aa a s than 50' at the most, telescope remains at vation of both stars, z and z' may be read vhich pass within 25 )e has a long diagonal )cus, and the micro- neridian. There is a jscope, and a vertical reading thfj level at allow for any change ;ope. le of the principle must be mp'- to construction 'he titude being aheady is found from a star ithin the field of view hich have nearly the For this is that their t in interval of time remain unchanged : ived for reading the n azimuth ; say, not mty. The meridian ained by transits of e chronometer time will culminate. The the meridian, ready id set for the mean altitude of the two st- r^ f 1,^ T ~ ' — w.re a. the calculated ins.an, oH ^t I„^f ""^ '"■"°™«=^ the level and micromerer r. '"^"•'nsit. Hethen reads ■ anner. I( after ^hfr, ''"'"'' '^" ''" 'he same purpose, which does no, a Lr fh '* ^'""'"'"^ '" ">« -|;3Cope and .he >^:CC:,Z:ZTClT-' ''' has several Sxed ver.ica ^ires 'o Z°, ■°.'- ■ '' "^"^"^ may be used for transits. ""^ 'ns.rument re»fv:tr'':ht:is': r"'-^ "" '^'^-^^ - 'he ^'ops for getting,' in, h'y T^""' "'''<' "«h 'wo a-idinglvel?or^ttgt:4''ve;-:f*'^"'^"''^'- .eleC:."^'" '"^ '"^'"-"' - ".ou„.ed like .he transit o^:^X!iz:^ tT' '•' ■""""> - '^^■■ U. S. Engir?ers It, ntf ? ^ ,''P'="" ™^°" °f 'he 'o ob,ai„ a sufficient-n r T """ " '' "f"" dMcult Which .he rcHnSraretct? ;:t ''^'^ °1 ^"-- "^ we have .o use .he ,m»n ""^'^V ^nown. As a rule veryvvellkno™ andruftl '7' ""r "'''"= "« -o' her of pairs ,o eiCL?::':^:'" °'"™ ^ '^'^^ "™- 'he raicr6„'e.er Ve^ aid T"' '" '"^ ^"' ^==™=^ - represented by ,«, ::ierh;ie: rX^ ■ 'C' \f """ .he s.ar.s apparent'. eSrd^lt It ufdtr ^"' -'°' I*' ' f •iBl 8o The Zenith Telescope. li i Let / be the equivalent in arc of the level reading, posi- tion when the reading of the north end of the level is the greater. Let r be the refraction. Then the true zenith distance of the southern star, or z, is: The quantity z^ +1)1^ is constant so long as the relation of the level and telescope is not changed. We have, there- fore, for the northern star, 3'=2o+'Wo -m'—l'+r Hence p,—s'^in'—m + l'-{-l+r—r' and the equation for the latitude previously given will become : ;=J id-\-d') + ^ (w'-m) + i il'-t-l) + ^ {r-r') TO FIND THE CORRECTION FOR LEVEL. Calling the readings of the north and south ends of the bubble M and s, and the inclinations at the observations of the north and south stars, expressed in divisions of the level, L' and L, we shall have L'= n L= n — s 2 2 and if D is the value of a division of the level in seconds of arc, we have r=v D /=L D and the correction for the level will be i (/'+0=i (L'+L) J^=''^-ii^A D 4 TO FIND THE VALUE OF A DIVISION OF THE LEVEL. Turn the telescope on a well-defined distant mark. Set the level to an extreme reading L, bisect it by the micrometer wire, and let the micrometer reading be M. Now move the telescope and level together by the tangent micrometer reading be M' Th f "'"■ ^""^ '=' the "■e .eve, in .„r„s o?.„e mi^ro.e^er'wm b°/ ^ '"''"'•' "' M-M' — L'— L arc wiJI be '"^ ^ °^ t'le level in seconds of -,. . . MLTER SCREW. no^r i^;tttTo„'';,t~- ofa circumpoiarsur -enith distance by the fo™„lce '" """"■ ''"Sk and cos (=cot S tan i Whence, knowinTtheTa's^Rt' . . error, we find the chronon^e er L; '?!"'' '=''™"<'««er CTt.on. Set the telescone fo* ,V u"'^ S'"««« "Ion. it upon the star .o TTjZ^\'T'' '"''^"« '• direct «. elongation, and l£cTkZ^T ">= •»« °f ^reat- note the time of bisection and ,^ m.crometer wire; ■nS^- As the star moves ver.t?,r''°""" '"" ''^"' «»d; often as possible whikit s ", r^M ""'' P^""=^ « '.. '.. '., &c., be the not d h"no ,°" °' ™*- Let "°n. '»,. «„ m, r ,h. °"""'- ""« of bisec. readings, « ','he mictometer ^T"™""'^ "'""'"eter greatest elongation (0, and T 7 "^f ""^ "^'a« of angu,ardis.a„ees;tbenthe,atVerl;e*?;„t^lt.rel-:;-«^ f!" •■==■>('-<,) cos * ,fg. ''"'•=™f'~;.)cos* -icrometer'.'Lr!" :ie'Vv:rh?' °' ^"™'°'■■°'■ "f «,) R=tg Therefore, subtracting (wig — m,) R=i^ — t, or R=-±^—''^ To correct for any change in the level reading, let /j and /, be the level readings corresponding to »i, and m,; then (/,— /j) D is the change required. The angular value of D is unknown ; but, since D=dR, the correction to be appHed to (?,— t,) is (/j— /,) dR; and (w,— Wi) R=»i— t, ± (/,—/,) dR or R=. 1,-13 A value of R is thus obtained for each of the observa- tions, and the mean of the results taken. This mean has then to be corrected for refraction, thus : From the tables find the change in refraction for i' at the zenith distance z. Let this change be dr ; then R dr will be the correc- tion to be subt cted from R. REDUCTION TO THE MERIDIAN. If a star has not been observed exactly on the meri- dian it may be taken when off it, and the observation re- duced. The following is one method of doing this. Keep- ing the instrument clamped in the meridian, the star is observed at a certain distance from the middle vertical thread and the time noted. This will give its hour angle, and if we denote this by t (in seconds of time) the reduc- tion is obtained by the formula i (15 t)» sin I" sin 2 d This is to be added to the observed zenith distance of a southern star, or subtracted from that of a northern one, and, in either case, half of it is to be added to the latitude! REFRACTION. When the zenith distances are small th. r • vanes as the tangent of the zenith distance. "'''''"" Let r=a tan z r'=afam' Thenr-r=«(tan2^tan2') sin (z -^') cos z cos z' fl sin i' cos' 2' =(2—2') nearly sufficiently delicate it mlv b/ *'%'^"''"'^ '='^'=fe is made ;ev^i..ei.t--tTst:::rnrer: must therefore be determined. ^^ "'"^' ^"^ tf ill CHAPTER IX. I'M f II ADDITIONAL METHODS OF FINDING THE LATtTUDE. TO FIND THE LATITUDE BY A SINGLE ALTITUDE TAKEN AT A KNOWN TIME. Here we have in the triangle P Z S the hour I angle P, the side Z S (go"— the objects alti- ' tude), and P S the polar distance. From these data we have to find V Z. From S draw S M perpendicular to P Z produced. Let H be the declination,

— 90° -,., Also ■:co,w--'j;tMS ''-' '''* ^^^-- ^rlvh cos P M : cos Z M ::cos P S : cos Z S or cos P M : sin (P M+tp) ::sin 8 : sin a Therefore sin (P M+^)=?l5A^L_M sin 8 (a) Equation (i) gives P M and (2) gives P M+

.» I Hi., m 'I il 86 Circum-Meridian Altitudes. Hence, in circular measure 1=^(1— p cos h-\-^ /)» sin» h tan a or in sexagesimal measure l=a—p cos h + .J /)» sin i " sin« h tan « This is the method given in the explanations at the end of the Nautical Almanac. To find the latitude we have only to take an altitude of Polaris, note the time (which will give us the sidereal time), and apply certain correc- tions as directed in the Almanac. FINDING THE LATITUDE BY CIRCUM-MERIDIAN ALTITUDES. When the latitude has been found by a single meridian altitude the result is only approximately true. It may, however, be obtained with great exactness by taking a number of altitudes of the sun or a star when within about a quarter of an hour of the meridian on either side of it. The altitudes may be taken with the sextant, reflecting circle, or theodolite, and the observations should follow each other quickly, and at about equal intervals of time. The watch error must be exactly known, and the time of each altitude noted. The mean of the altitudes is taken, but the hour angle for each must be obtained separately. In the case of the sun this is done by cor- recting the observed times for watch error and subtracting them from the mean time of apparent noon. If a star is used the mean time corresponding to its R> A. will, of course, give the hour angles— The formula is L2itit\ide—go'—a±d—x' Where a is the mean of the altitudes, d the declination of the object (negative if south), and x" a quantity equal to 2 sm' — Xcos.approx. lat. x cos. dec'n. x sec. alt. ; h being sm I the hour angle. Circum. Meridian Altitudes, To prove the formula ~ "" CZr'"'~''- ""■"' '"= '••"■•'"'''■■ - .approximately 2 sin "A and .t"= 2 ^ cos / cos d 5in I COS a We have now to determine the small quantity .- ^H W sm PZ sin PS cos ZPS^cos ZS-co Pz W ^^ 01; cos / cos d cos A=.sin «-sin / sin J '^ ' ^^' .*. COS / cos d (I— cos A^ => cr, , '^ v^ LOS n) ^ —sin rt 4- cos (/__^^ - -sina-|-sin(rt-|-;t) .'.2 cos/ cos rf sin » ~~ 2 Therefore, approximately "2 sin-^ cos (« + ^ A -« 2 ^ ^ . iA 2 sm — y'=- i^v^°^ ' cosrf si" I" ■ cos^a^ - «^ is, Of course, negative if south. The value of the expression 1!;'!1t sin I ,, ,. =ui I ^^"own as the reduction to the meridian") ,<= f ^ r angle from a table and th! I ?"^ ^^^ "^^^ ^^our to calculate ;.- with ""'"" °^^" ^^^ ^^^^es taken whSTatbTe^a^pr^a^ " ^ ''^^^ ^^^ ^-^^-^e of the mean of ten X rfc^^^ -ed with a powerful theod^l^ w^ 3Vr20-"^^Th°-^- / I I 88 Circum-Mendtan Altitudes. when corrected for refraction, parallax, and semi-diameter, gave 40 14 3i".55 as the true mean altitude of the sun's centre. The sun's declination was 19° 53' 45",8 south The mean of the values of the reduction for the observed hour angles, as taken from the table, was i6".26, and the calculated value of x was 17". 36. go* o' o" ••40 14 31-55 Altitude 49 45 28.45 Declination... ig 53 45.80 2g 51 42.65 17-36 Latitude_2g 51 25. 2g Strictly speaking, a further correction ought to be mkde for the change in the sun's declination during the obser- vations. In the case of a star we must add 0.0023715 to the log otx" to correct the hour angle for the difference between the sidereal and mean time intervals; for the star moves faster than the sun, and therefore gives a larger hour angle for the same time. Additional accuracy is obtained by taking half the ob- servations east of the meridian, and half west of it, the mtervalsof time between the successive observations being made as nearly equal as possible. The hour angle changes its sign after the meridian passage of the object. A^'^ ^/rt^ / Cm. 7' 31 \ CHAPTER X. m INTERPOLATION. METHODS OF FWDWO THB LOm.TVDE. and longitude of 'he 'iLl '?'7°'="' ^°' ">= 'ocal ttae va'atrrsfaZ?oV!;v::t!:h-:i^ '^ "-" accurate result. *° °^*^^" a very a cena. ,ea. a. a pr rLSd?«X'.-r " For Greenwich n,ean noon we find in the Aln,a„ac , ^^' '■ "° l^5".t::;:JK5 "ow, at apparent noon at the olarf if , -ii i. apparent time at Greenwich, anHrke ,he .-^ '•"• ' ;: "--*"='• - "■= "-"^^ vaSno: /onhretrn.:' Th.s vanation is X3..305, which m„„iphed by 4 g,Ve, •I I * l;n : !'i 90 Interpolation. 53"-22 to be subtracted from the de- dination of 2d January — 22° 57' l6".2 ff3"-22 22 56 22.g8=required dcc'n. Variation at a p.u 14-35 13-21 12) 1. 14 -095 13.21 13-305= -^ 53.22 INTERPOLATION BY SECOND DIFFERENCES. The differences between the successive values of the quantities given in the Nautical Almanac as functions of the time are called the;?n-/ differences; the differences be- tween these successive differences are called second Merences; the differences of the second differences are third differences, and so on. In simple interpolation we assume the function to vary uniformly ; that is, that the hrst difference ,s constant, and therefore that there is no econd difference. If this is not the case simple interpo- lation will give an incorrect result, and we must resort to interpolation by second differences, in which we take into account the variation in the first difference, but assume Its variation to be constant and that there is no third uinerence. The formula employed is f {a + k)=f {a) + \ k-\-B k^ where A is half the srm of two consecutive first differ- ences and B is half their difference. It is thus derived: We have by Taylor's Theorem / {X + //) =f {x)+Mi + H/;2 + &c. (A) and if A is small compared with x the successive terms ol the series grow rapidly less. ^Varlsiion at 9 p.u Jnterpolation. sum d il't: ^""^"-"^O^J I'-on./Cv, i„ which i, is "a . fore we know.hedife no^^S-A'^ 7 "'.T," ""="■ obfar r;r '^f; ,1':--- -- ^e „e«,ectrd we Z follows: ^ '■*"*'' "'"=■■'= * '= '"''^ "mn I, as /(« + i)=/(«)4.A4.B \A fi^+k)=f{a)+Ak + Bk^ ;2 Subtracting equation (i) from (2) we get and subtracting (2) from (3) ^^l+I ••• A=J (A'-f A) .'. substituting in (4) ^^The method can he he.ter understood from an example Numhera. 365 366 ,367 Lo(j, 5622929 5634811 5646661 lit Diff^r'ce. 3d Differ'ce. I1882 I1850 i }. —32 ill, 92 Interpolation. Here k{s^,A =ii866. and B=-i6. 563481 I 4746 "«66XA 47464 ""¥ X (A) ^=—3. nearly. 3639554 The tables give the log. as 3639555, If the second difference had been neelected-/ . .f U«< Difference. Log. COS. 89 32=7.9108793 Log. COS. 89 33=7-8950854 Log. COS. 8q 34=7.87860 ^^ Slid Difference. ' —157939 , ^ — 163901 -5962 Here we have to subtract ^^ x hairthT^m of the^ ist 4^r6Tiir' ''^'' ""'''' ''' --^ ^^ff-nceV:: •'• ^^S' COS. 890 33 ' i5"= 7.8910438. TO FIND THE GREENWICH TIME CORRESPONDING TO A GIVEN RIGHT ASCENSION OF THE MOON ON A GIVEN D.^V. Let T'=the Greenwich time corresponding to ^he given right ascension a' ^ T=the Greenwich hour preceding T' and correspond- ing to the right ascension a ^ ""^ dme ?''"'' °^ ^' ^' '" °"' ""'""'" ^* *'^« Then we shall have, approximately, «'— a T— T = A a I interpolation. instant of the internal T'-l .^^tlTr^'^'* *^" "^'^^"^ have • ^'^" ^'"s A , and we shall T— T= a — a T and T are in minutes. \ w=i2\ n=^%=:L. a^rl If we denote the co-efficients of a, b, c, &c. In (c^ L a a, C, &c., we have w j- ^, 0= -I- 28^478.04, A=« 36^97, B=A 4b-79. C=B 6=— 11 — I 2 n — 2 1..74. D=c -^-=::3 4 o,.66, E^d"~4 5 =~h Bb= + '"l~isr^fl"» Ec=- 4^62 o*.3o o*.o7 Og.02 Moon'oR.A.onMarch5,6h, or F^^^ =22" I2«°- 56 .74 TO FIND THE LONGITUDE BY TRANSITS OF MOON- CULMINATING STARS. This is a simple and easy way of finding the longitude when the mend.an line is known, though not a very ac! curate one; for an error of one second in an observed trans t may throw the longitude out as much as half a minute m time, or 7I minutes in arc. It is, however a raethod that may be useful to a surveyor, since all he waTch" oT " '' " '''""* '^'°'°"'^ ^"^ ^" -binary watch. Of course, a portable transit instrument is to be preferred, if available. The instrument is set up in the plane of the meridian, i I! a -OS.C6 LoHiiitud^ by Moon-Culminations. 4^62 o*.3o o».o7 0,.02 — 95 know either Greenwich or loca -.in buM "",""7 '" is .educed IZ'Ct totiraUil""" ^""-" '"- .ear.t':siT:nr:sr;/r^-tf''^ moon and of certain suitable stars calllH^."^ °^ "'" ins" stars ,- also the rate of chaT^e J LurTarttr'" of transit) of the moon's R. A As th^i '" rapidjy thronrf, the stars from w^st toes, T™ T'' thatfa station not c„ the me"idan „, r^" '' ."'''\"' -nterval between the two transi."tli; be dSeTeTto tW at Greenw,ch ,■ and, the moon's rate of mottoTl.l bemg known a sin.ple proportion will ( H .sta, o^" near the meridian of Greenwich) rive c iff , ..m. between the station and Griem ch ,d ,!""=\°' longitude. If the station is far fro ,, Ih T *" Greenwich a correction will have,,^ I T'j''"" °' change in the rate of chanJ'ofTe 1 .s^R^V^^L^ NXf.z„:i-Xttre^ time between the transits! ^^'°^' '"*^^^^' ^^ An example will best illustrate the method :- At Kingston, Canada, on the 24th Februarv tS«. .u transits of the star v Tauri and of fh ^'^'^f^' ^^^2, the were observed at 6h. om ,3 ' [^,\TT '""'* '"' tively, mean time. Differe^fct ^G eeonds";.'-';^"^'"- sidereal units. . 4" t,tconcis, 01 465.12 jn -;:ll I m I hfim m\ I 1 ^ ^ongitude by Moon-Culminations. Greenwich Transits fL^^"'^''^'^- ^^m. 163.62 I Moon I. ..4 7 57 .44 Difference in sidereal time= iim. iqs.iS Add interval at Kingston= * 46 .12 Total change of moon's R.A= 12m 5s'!^=725s.3 By interpolation by second differences the variation of he moon's R.A. per hour at Kingston at the time of transit was found to be j^gs .23 At Greenwich it was 1^2 .68 2)284 -91 Mean rate of variation ^^5 j^gXj.h. = 5h.09i6 5h. 5m. 29s. 76 west longitude. It should be noted that in this case the moon was west ot the star at transit at Greenwich and east of it at Kmgston, having passed it in the interval. The following is a specimen of the part of the Nautical Almanac relatmg to moon-culminating stars. T tron fi:one othe one. ever, Moon-Culininating Stars. 25S.3 ition of time of ^'itiide. IS west f it at lutical 97 gone wrong. The ;„.,. ^°^ chronometer has ™e. An error of 30" i„ "ad^f 1""°" f " "^O- simple -- - -r . .on,.,-- - -tTai:;:: 4. ih !l iiSlj 98 Longitude by Lunar Distances. The moon moves amongst the stars from west to east at the rate of about 13' a day. Its anfjular distance from the sun or certain stars may therefore he taken as an in- dication of Greenwich mean time at any instant— the moon being in fact made use of as a clock in the sky to show Greenwich mean time at the instant of observation. The local mean time being also supposed to be known, we have the requisite data for determining the longitude of a station. In the Nautical Almi^nac are given for every 3d hour of G.M.T. the angular distances of the apparent centre of the moon from f.ie sun, the larger planets, and certain stars, as they would appear from the centre of the earth. When a lunar distance has been observed it has to be reduced to the centre of the earth by clearing it of the effects of parallax and refraction, and the numbers in the Nautical Almanac give th- exaci. Greenwich mean time at which the objects would have the same distance. It is to be noted that, though the combined effect of parallax and refraction increases the apparent altitude of the sun or a star, in the case of the moon, owing to its near- ness to the earth, the parallax is greater than the refrac- tion, and the altitude is lessened. Three observations are required— one of the lunar dis- tance, one of the moon's altitude, and one of the other object's altitude. The altitudes need not be observed with the same care as the distance. The clock time of the observations must also be noted. The sextant is the instrument generally used. All the observations can be taken by one observer, but it is better to have three or four. If one of the objects is at a proper distance from the meri- dian the local mean time can be inferred from its altitude. If it is too near the meridian the watch error must be found by an altitude taken either before or after the lunar observation. l^ongitude by Lunar Distances. P —^ _ 99 J.,.Ji.. .,.«», ,„..,.„„ .,,„„„„,„„ ^^ „,„,„,„^.,.^„„^ 4tb Mean Totalt. If there is only servations in iC f^^n °^'^'"''^'" /' ''^ «est to take the ob- watch. IS at of "^ °'^''"' "°^'"^ *'"-^ ^'"^- by a ^»- 1st, aJt. of sun, star or planet- 2fi olf ^r 3d, any odd number of distances ^thlu f "'°°"' alt. of sun, star, or nlanlt T ^ ^1' ' "^"""""^ 5th. tances and of the" fw" tII T "f " "' ^^" '^'■^■ the mean of the t 'ne •- 7 't "'' *'''^ "'*'*"^^''' ^^ ence of time o. If I i ' Ti '^' P'-^PO" tion-differ- ^i'-ofxJt ,ndn " V,''^- °^^'*^--^'^''^- between which is to e ad ed r K ''"'^ •' ^ '^"^^^ ""^^er responding to that mean ^' ''"^''' ^'^ ^°'-- -u^aratd^t^^ r' ^^^^^ '""^^ ""^ ^--^^^ - aCded to the distanc ,'' -^"d'ameter of the moon Ti^e lunar dstanceTa,?,'^''' '^^ ^'^^^"^'^ °f ^^^ "Centre, of Parallaxt^d™^^^^^^ ^^ ^^ ^^--'^ °^ the effects TO ^^TERMmE THE LUNAR DISTANCE CLEARED OF PARALLAX AND REFRACTION. and 7c fv. ^^ *''' ^^^^'•^^'•'s zenith, Zm and Z. the vertical circles in which the 7„7^"^^.t--e situated at the instl of ohservafon. Let m and . be their observed places, M and S their places after correction for parallax and refrac tion : then Z,„,.Z., and,,, are found b, observation, andZM and ZSareobtaLd by correcting the observations. Theob- ^^ ,, 11 ■"■it Too LongiHuie by Lunar Distan ces. ject o^ the ralculation is to det ermine M S. Now, as the angle Z is com and M ZS, we can find Z fron „„^. all the sides are known. Next, in triangl nion to the trianf,'lcs mZs 1 the trianf,'le mZs in which are known M Z, Z S .. e MZS there and the included an>,dc Z, from eared lunar dis- which M S can be found. M S is the cl tance. The numerical work of this process is tedious. The cleared distance having been obtained we proceed in accordance with the rules given in the N.A. The Greenwich mean time corresponding to'the cleared distance can be found either by a simple proportion or by proportional logs. ^ It admits of proof that if D is the moon's semi-diameter as seen from the centre of the earth (given in N.A) D' . s semj-dmmeter as seen by a spectator in whose .enith t IS, D Its semi-d.ameter as seen at a point where itsalti- tude is (t, then D'— D=(D'— D) sin a, very nearly. For details of the methods of hn.ling differences of ong. ude by the transportation of chronometers, and by the electric telegraph, vide Chauvenet or Loomi. <-HAI>TKR XI. The amplitude is tho anWo tir.t H . vertical circio tliroM-^h -,n ?. *''''^""-' l''='"« "f the of the primo vortical '"'' '"''''^■^^ ^^■'■^'' *'"• I^'^-^'-" ,,^'.;;^;^^^^^'-'''-orti..so„ti, ■•^-^.t Ml west points „ftl,clH.nV,.„ 'eavcniy body. Suppose H to be HncWH is the amplittuie (a) polar distance (go^-^; u l' H N P ^']' " ^ ^^" "^^-^'^ ^ 's the hour angle. H P N- x8o^ /'^ ';"^''" ^'^°' '^• Hence; « , n i^ JN i8o -^, and N Hr=^^-a. sin a= cos^= sec f sin . ^ -tan f tan ^J •^^"=r,::^^-i:e.„...„,. I n l# -^ 1 -fif I' I I02 Parallax. TO FIND THE EQUATORIAL HORIZONTAL PARALLAX OF A HEAVENLV BODY AT A GIVEN DISTANCE FROM THE CENTRE OF THE EARTH. Referring to the figure in the next article, if A is the observer s position H' will be the apparent position of the heavenly body, and if C be the centre of the earth the equatorial horizontal parallax will be the angle H' Desit^ nating A C by . A H' or C H by ., and the^ paraila^;i sin/=^ TO FIND THE PARALLAX IN ALTITUDE, THE EARTH BEING REGARDED AS A SPHERE. In Fig. 29 A is the observ- er's position, Z the zenith, C H f the rational horizon, A H' the sensible horizon, and S the heavenly body. Let p be the horizontal parallax (H'), p' the parallax in altitude (S), h the altitude (S A H'), and d the distance of the heavenly body (S C). From the trianTle s'a C we have _ .l'":^_ _ __^n_S _ A C sin Z A S ~ sin S"Tc~" SC ^ =-^=sin/ or smp =cos h sin p The angles/ and p' being (except in the case of th-^ .moon) very small, we may substitute them for their sines,' and the equation becomes P'=p cos h STAR CATALOGUES. If we want to find the position of a star not included amongst the small number (197) given in the Nautical Almanac we must refer to a star catalogue. In these Star Catalagues. 103 right ascensions and deciina^onf, "^ """' "PP"""" co-ordinatesare always 'n^C sfLT" '''"• ^""'^ tion, and aberration which r! 7 i' f ' ^ P'^'='^«='°".nuta- Position ; 2ndly bv Ihl it -"■ "''P"™' <='«"Kes of selves atnon^st^'ea^h' b ri'r;":.:: ""^ ^'? *"=- are referred to a mp^n « / catalogues the stars some assumed :poTh ZZ '"'. ' "^^" ^^"'"^ ^^ called its mean pice at tha r .' '*'' '" ^^^^^^^^ is to the true equator and t ' '^'' "^" ^^^^ ^^^^^^ed that in whicl^Tfs ^r pXsr:°^ 'l^ ^^ P^-^; and its ./A,.,.^ place. TheTn" „ it f '''''' ^" "°^^°" found from that of tl ^ ^.^ ^' """^ ''""^ ^^" be precession and ?heprooe'''''°T '^ ^^^P^^'"^ the elapsed .incetLeLrh of thrf," " *^'' ^'"^^ ^^^^ ^^^ then be found "v c^^^^^^^ and, lastly, the aVparen pface sTu^^t" '"""^^^'°" ' true place for aberration. ^^ correcting the The most noteworthy star ^o+^i Association CataloguefB a C 'T" "" ""^ "^""^h the Greenwich catalog fs' Lahnde"" * ''^'^ ='='■■=• 50,000, Struve-s, ArgeLder s!fc &c ""'""""^ "^"'^ It is"I"™"" ''""'™^ °^ co.«.,«,e,. ™ine:batX"re;\,-;,rrri,r"?"°'"^ '<> ^«- in .he ,nanti.ies co:;;.:d "rth ^ tAr'" """"- are very small the simpler differential ™"al»ns used. The-most nseW differentia fL^",™' "='>' ''" spherical triangles, are ded!:rd1's1ofw.''' ^^ '^^"^^ We have the fundamental equations: cos a=cos 6 cos c + sin b sin c co., A ! ll i- I ■ ■] JJuJ I04 Differential Variations. Differentiating the first equation of this^T;^^;^^ changing signs, we have ^ ^' '^""^ sin a da^sin b cos cdb + cos 6 sin crf.-cos 6 sin c cos A db __ -sm b cos c cos A ^c-f sin 6 sin c sin A ./ A -(sin 6 cos c-cosft sin c cos A) ^6 + (cos 6 sin c-sin ^, cos c cos A) ^c + sin b sin c sin A ./ A =sin a cos C rf6 + sin a poq R ^Z . • r. • . ^ ^^ ^ ««-^sln a cos Ji rfc + sin b sin c sin A da or ^«=cos C ^6 + cos Brfc + sin b sin c-^i" 4,/A =cos C db + cos B ./.+sin 6 sin C rf A ''" " or^«-cos C rfft-cos B ./c=sin b sin C dA Similarly we obtain ^ —cos C da-\-db~cos A f/c=sin c ^\u A / u ' (2) -CO. I3^„_c<,s A rf*+*=sh „ sin B i'c J From these, by eliminatinK da, we obtain • and by eliminating db from these : ^ " ^' ^ <-J «in«sinBrfc^cos6^A+cos«^B+.;c (4) If we eliminate d A from (3) we get cos 6 sin C db — cos c c;in n ^. ^ cos c sm B ^c-=sin c cos B cf B —sin 6 cos C ^ C and, by dividing this equation by sin b sin r equivalent sin c sin B, we have ^ °' '^^ cot 6 rf6-cot c rfc^cot B rf B-cot C ^<^ (,) anfpuT ^"'"^'^' *^'^ the astronomical triangle P Z S. B If ,«Zf °-^ Then the first equations of (2) and (3) give cos /^t~sTn I d r""" '^ ^"^^ ^' ^ ^+ -°^ ^ ^ ^) ... Differential Variations. Its 105 (which are derived dir.^ii r ''" ^ ''" -^ «ons(l)), whe„c,AndTaLff°"; 'J? f""'*'""^"'*! equa- '' 2. and rf ^ resSely '''"^"'^■•""""-"ors^. In a similar manner we obtain -sin c/z^- ~sir/i/"'" ? cos * rf (_^o, 2i'„ , ^v;.ich de.er.nL' 'i e^ TrLrj ^ 1'7?J ^ ^"' ^^^f '^' "f C and Z comp„.ed hy .h:ir1„'„ S'd' Ltd t ?'"■=> J -n C sin Z^-cos I sin f~''" *" '=°= " "' ' (9) ^'nnd-y^Xrcreir^'^" "-^ "-^ -a« erL.*, anjie S, or parallactic angle r Ihe J ' ,''"""^«<'". ? 'he and t the star's houranglf.) "' ^'""* ''''^'ance, TO FIND THE CORRECTION FOR SM.,. ™e altitudes whe/^oTkc ^„'f «"aut,es '" eOU.. ,„,T..ES OF . FtXEO ST.r "" ■^e reSio^f Xtt'lt ^ l^ro^ '"^ "-P-"- apparent altitudes will no. give eori f''™""''^. equal find the change At in th» ^ ^ '™<: altitudes. To change A « in tte altitude » I ""«'" P-'od-ced by T 'he equation. "''' ' ""^ ''^^'^ ""'y to differentfate Sin,=sin,,sin*+cos^cosicos; Resardmg, and. as constant: whence -"e.„';:i:r:nro7ar:„n;-' time. ^'^c, and a^ m seconds of '^'^-f'^de at the west observation is the greater h. f t Ill io6 Effect of Errors. From the above equation A a the hour angle is increased by At, and the middle time is increased by — , which is therefore the correction for the difference of altitudes its value is A« cos a 30 cos f cos d sin t If A is the azimuth of the object, we have Sin A =.?^A"J cos a and the formula may be written A« 30 cos

nZ is i„a„i,e; therefore fJhi ,'"'''= f"™" "se we ean s.i,, ,e. ^ood"::::; ^ obs J vi"' :.r :"":^ prime vertical. ^"^^iving stars near the If ^C-o and daxi,„,™ whe^ „ Z-,> P"™/"""' <-"« stars, those „e.ir,l,„ . '' '""^ ""'"• ""iff^fent In I r, ''•"'"°'' "' "'^ best to observe. 'voidiot a,'trdrs''.:obf ^" ': "^"^=^^>' » -<<" •» the prime ver caT ' In th ""', "' " '"^'='"« f™™ v'iU aiTeet the H. u "°' """" "™'' '" *= data served on succe";°"""°u- '"" " "'^ ''" '^ "b- dian at about ,h's»^^' °" '^t '^"" =''^' "' "-e meri. accurately Ob ai'rr.'T""' '"" "'"""'^ ""' ""' "e certain. ' ""«'' "" ^""^1 .,„. „i|i 5, „„. n.eridt„:T„r:r;he°'""^',''°"' ^^=' ^-'^ -=< °f "-a errors 2 i, a^j % T' *""'=" f"™ "• instant ^ithopp^^esi^s ^'h^ ^'" ""; ^'"'^™l-of*, but too large and ,.: "^"^^ °"^ -^'^ek correction will be amount a'nd"hei 1!' n'r T"' """ ''^ ">= ^^me at the tLe ofte 3^:1^*::^'-^ ""-^- CIRCUM-MERIDIAN ALTITUDES. r from r"""^' ^°' ^"^^"^ ^*^^ '"^^idian zenith distance C from a crcum-meridian zenith distance C is .' il r t IM io8 The probable error. where A =^°^^ ^"^J m D have sin ^ and ;« this. fferentiat , since dip=d^ + dd df z sin 2 sin i' and reirar ■egaiding A as constant, we ,irin A sin t "C + iio - — „- 15 dt sni I "^ The errors ./C and dS affect the resultin,^ latitude i,y their whole amount. The coefficient of dt has opposite signs for east and west hour an.des ; therefore, if ob;e - va .ons are taken of a number of pairs of equal altitudes east and west of the meridian, a small constant error i„ he hour angles (or clock correction) will he eliminated taking the same number of observations at each side of the meridian, and at nearly equal intervals of time. An error in the assumed latitude which affects A is fo!,"n;rh''i V'^''"''"^ ''"' computation with the latitude tound by the hrst one. THE PROBABLE ERROR. To give an idea of what is meant by the term "proba. le erro.r we will suppose a rifleman to have fiVed a large number of bullets at a target at the same range and with equal care in aiming, and that, on examining the target it is found that half of them have struck within three feet of the centre, and half outside that radius • then it may be assumed, «^^,wf, that the chances of any one shot hitting within 3 feet of the centre are even-in other words, that it is an even chance whether or not the bullet will strike within that distance or not. And this distance may be taken as the probable error of any one Now, if we make a series of independent but equally careful measurements of a given quantity, such as an ill The probable error. or a base line, they will all differ angle closeness of the af,Teement employed and th 109 more or less, the <'';P^'nding on the instruments decide what value is to be tlie correct one— in oth f care exercised; and tl le proble m is to taken as the most likely to be probable error. '' '^°"^' ^^ ^^^^^ ^^'"^ ^"lallest If «.. <'.. .,.&c.. are thedifferent measurements,,, their number, and m their mean, then w .«' + «« + &c. + «^^ ('"-".) will i,e „,;, r !^' ^';--.'' <"'-'.). &c. calkiltlie "residuah" A.. , ■*"" 'luantitjes are «c. is a mmimum. ^ ^i) .('«~«3)», Hv^^o;;;;;:^:;!!;:^^''^"^^^"^^^^--'"^ be nearest the tr^^ T ifv^^ "''^'^ '^ "^^'^ ^° the mean is ^ ""^ ^'^^ probable error of n - -- ''0.674489 And the probable *>rrr.r ^t ^^^ p-auee.Loth;™:;:2;;;:;,7--"' ''^ "■= tl.at the value taken is wMi„ V, .f' """ '=''^"" without regard to sC TJ^uZi"' '"' *"■"■ number of measurements of Th r ^ "'^''" "' =■ P^baWe error. i.isrL::hT„:J;ttt'' ' f~'/'^ lies between /— i and /+i. ^ ^^^' ^^lue e-nX^'^t^sf/rrtTT-'r^^^ cates the relative value to b^ assiLejrth"'"','"'"- regards precision. assigned to the results as 1 ): i ;^ •^ce.1^7^ ^1-e ^i'fe ^^ Ji^ ^^ '/ *-tc« /^>/^ ^-V -^^ *a^5i^->-< «^t^^SL r' ;%.. .;*ll #v,*rr II S i IIO The probable error. The formula tor the weight IS zK'^ M3 pt~a,)» + {m~a^)''T&^~;r{;;iira;jrf (2) 111/. '^T/ Probable .r.-„r- °:ff^936 "/ "'»Ci3) V weight •nri" m"' ""^ "'°''''" ™"'=' inversely as the square of the JZLT\- r^-'l-P-P^'Vofthe sumofthesquares of (he residuals be.ng a minimum in the case ofthe mean th,s method ,s often called the "method of least squares": As a simple example ofthe calculation of tie probable error we w,ll take a side of a triangle forming part bf a nangulat,on carried out near Kingston in rSs!! FoVr letHnTtirrrsXr:.-- -- - ---- ^- 1 060. 1 yards 1060.9 1060.6 1060.4 4)4242.0 I 2 3 4 Mean= 1060.5 Here the squares ofthe residuals, in tenths of yards, are I. 2. 3. 4- 16 16 I I Total 34 And the probable error ofthe mean is 1/34 ^~ X 0.674489x3-6 inches. — ■ 3 . 546 inches. a u k n. e( Pi w eq Part II. GEODESY. CHAPTER I. :•■ 1 THE FIGURE OF THE EARTH. an s„ve,i„, operiL of :::;;';sf .Lm- ure of the earth has to be taken mf , ^^ ^^■ lu ue taken into consideration minor a;as— the nnlnr • \ ^^ '^°""^ >ts aiil f li 1X2 Figure of the earth. ments of portions of meridianal arcs A litf lo nr. ■ i .ion win show that if .„„ curvatur^:. al';" i, .7: r U e i.pt.cal.and therefore decroa.i„s t..wa„ls .l,c pole ,1 e le..g.l, on the cartl.'s .surface of a .lepeo of latitude nut be greater ,n high than i„ ,„„ |„,i,„,|^^ ^^' <; " '^' an.l B are two point, on a „,eri,lian near the e ; t„f and i. and IJ two ponits on a meridian in a hi..|i l.til,,,!, lances Ali ami C D are measured on the LTound A U will be found to be less than C D. This h\s tuallv been done at various parts of the earth's surface-Lan^ land, Peru Franee, Russia, (where an arc of over ..-^vt from if. '"T'"'' ^ ^''^^^ "^^V accurately, and from t to connect by means of a chain of trian.^u Vitio two distant stations which areas nearly as pSt ^n he same mer.d,an. This being done we Jn calcula e the actual distance from one of the stations to the po^t where a perpendicular drawn to the meridian of that s a Sn "" ;'r *'" ^^^*'°" '"^^^^ *he meridian The latitudes of the two stations are found by very carefr^ astronomical observations, and their differLce,'aken n connection with the calculated distance on the meridian gives the curvature of the arc, since the radius of cu^va latitude fn^'T'' ^"^""' '^"'^^^^ ^y '^'^ ^'ff— of atitude m circular measure. There is, however, one ource of error ,n determinations of this kind. In finding the latitudes of stations we are in general dependent on the direction of the plumb line; and should there as often happens, be a local abnormal deviation of ^he lat er from the true perpendicular, the resulting latitude wi 1 be erroneous. This was proved manv years ago by taking the latitudes of two stations on opposite sides of a mountain in Perthshire, and measuring t'he true hor l^ tal distance between them, when it was found that the . figure ofihe Earth, difference of the laflf iW<.c •~^~~^ " ' sWci of it. The i„r ,„ ■' ,1" ""■" '™ "' "PP"site ■he s..i„„s, as:.:::::::;r „ ;t:::r:fi'"''"''^ °' wer> as follows : ^ " °' observations, Betvveeen A and B... o-;" « " 1^ and c.....::::'^'-^^ whfe the actual differenceq n= f 4 u were: "ineiences, as found by triangulation, Between A, „dB.......,..^,,, 3".:' "■'= '~' °''"^ «"-ayas, whe" T;rr,„rr3 eartVs cruse beire,-„j'H '",""'' P""°"= °f ">e ase. The ac,„T aC „ oTr °[''""'' "■^" "^ aver, be ..scereained as foZs 'a reTo" ^f"'"" '"^'^ symmetricallygroupci ro„nH i? I "" ^'at.ons anJ lonsitude of ea '1, '\T'''°""''""^ theia.itude tioa.. The acVu^r J J ■"" ^ ='^'""°"'i"l observa- "ation from ea h If hr„H "1, '■"""'"' °^ '"= <=<=""al 'alion, its Ja.itude andtn ;'? ""*■' '"'"™ ''J' '"angu- •hat of each of th *• ""^ '"' "kulated from the result 1; b!"";" ""'"^'^'y- and the mean of the errors caused by putr,"";"'' "" '™'h. -"ce batons will probaWvl ? deviations at the various latitude and bngftud'e o "he":' T'^ °"'^^- '"''' """ par- witb the^a.i:uditv;:-idrrsT. 114 Figure of the Earth. Ki, m Mi 'i I astronomical observations will give the deviation of the plumb line. Ifrt and b are the semi-major and semi-minor axes of an ellipse, the distance of the centre from either focus is y'flS-Z^s, and this quantity divided by a is called the "eccentricity." This is {generally written e. The quantity u — b. is called the "compression" or "ellipticity," iind is denoted by c. The latest calculations mp.ke the com- pression of the earth about jJ, ,, the ratio of the semi- axes being believed to be 292 to 293. The true measure of the compression is the difference of the semi-axes divided by the mean radius of curvature of the spheroid. The equator has also been found to be elliptical, its major axis being about j©oo yards longer than its minor axis. It should be noted that the expression e has different meanings in different books. English writers occasion- ally employ it for the compression or ellipticity, while in American books it is used in the same sense as here, namely, for the eccentricity. Even in different chapters of the same work the letter e is often used both for the compression and the eccentricity. The accompanying 1 figure represents a [ section of the earth. PP' is the polar axis,| QE an equatorial di- ameter, C the centre, I F a focus of the ellipse, A a point on the surface, A T a tangent at A, and I Z A O perpendicular P^g- 3o. to A T. Z' is the geocentric zenith, and Z' C E' is its declination. The latter is called ih.& geocentric or rechiced lati- itid is FifTure of the Earth. "5 /MrftfofA. ZO'E'isthe aph C F» CE» a _ PC« "'^ C E2 7 A y " '•' "'^&'^'^K"'/""t«^ o>'as/m«owtcrt/ latitude, Z A Z or C A O is called the r.^«r//o« of the latitude. It IS evident that the geocentric is always less than the geographical latitude. LetCE:=«. CV=b. Let c be the compression and « the eccentricity. — inL- _ a , CF CF CE P 1 Pii;3— PC C E» That is, c2..= i -=.^i— (i_c)2 or, c-= i/gc^^ (I) TO FIND THE REDUCTION OF THE LATITUDE. Taking the centre of the ellipse as the origin of axes the equation of the ellipse will be 12 ^ hi ^ Let sin i" TO KIND THE RADIUS OF CURVATURE OF THE TERRESTRIAL MERIDIAN, FOR A GIVEN LATITUDE. Denote this radius by r". We have, from the Differential Calculus, ;?.= -^ I + my dx' ex Cif-i/' fr ^«^ / ii8 Figure of the Earth. From the equation to the ellipse we have dx d»y 6» X T whence R . {a* y^-\-b*' £2) ^ a< h* Observing that b- = «3 (i_,a), ^e find, by substitut- ing the values of ,vand^ in terms of ^ (page 115.) ^(^— ^') _..^/~-' -(6) R (t — g2 sin2 KrVefson of this is that the vertical plane at A passing through B will not I MA/ Figure of the Earth. "9 coincide with the vertical plane at B passing^d^i^^ii^ These two planes will, of course, intersect at A and B n *''^;^J"t^^sections with the surface of the spheroid will be different curves, and will enclose between them a space In addition to these two lines and the geodesic line there will also be what is known as the line of align. menioUhet^o points-that is the line on every point of which the line of sight of the telescope of a theodolite in perfect adjustment and truly levelled would, when directed on one station, intersect the other on the tele- scope being turned over. ■'p. I m I- '3 CHAPTER II. OEODETICAL OPERATIONS. The methods adopted in the old world for inappinff large tracts of country have been reversed in America Instead of startmg from carefully measured bases, and carrying out chains of triangulation connecting various prmcipal pomts in such a manner that the relative positions of the latter with respect to each other may be ascertamed within a few inches, though several hundred nrles apart, the system pursued (if we except the U S Loast burvey and some other triangulations) h;is been to take certain meridians and parallels of latitude inter- secting each other ; to trace and mark out these meridians and parallels on the ground ; to divide the figures enclosed by them into blocks or ''checks ;" and to further subdivide the latter into townships, sections, and quarter sections. Although the method of triangulation is incom- parably the most accurate, the American plan has the advantage of rapidity and cheapness. As the latter is very simple, and is fully explained in the Canadian Gov- ernment Manual of Survey, it will not be further touched upon here. At the commencement of a triangulation a piece of toierab.y level ground having been selected, a base line, Triangulatiott. 121 generally from k '•q in tth'Ipc ]Z^^- ~~ ~~ very greatest care LTr ^' \\measured with the chain nf ? ? ' • ^'°'" '* ^'^^^^ a network or a "sfena " if '' '° "'■"'™ ^<'''='=«=d points or at «o k /y^^Sonmetry. The instrument is then nlaced -en .00 .i,es andin one inst.JZt7 l^l^h: The bngest side ,„ the British triangulation was „ The s,des of the secondary triangles are from ab^m , M 20 m,Ies, and those of the tertiary trianglesTve ortf. The larger triangles should be as nearly equilateral =. circumstances admit of. The reason for h/^ 1 is that with this form smaH errors "n*. * *"" '° of. heir angles wil, have a ^Zl'lZ Z'^ZT '^^.Sfo:ey.t:■^ ^-^ «-^'-.e ctd The original base has to be reduced to the level of fh. sea-that ,s, the true distance between thenZt u verticals through its ends intersect the s^aUTmJ^b: I:'' 122 Tnangulation. 11 ascertained. The exact geographical position of one end and the azimuth of the other with respect to it. must of course be known. The angles of all the principal triandes must be measured with the greatest exactness that the best instruments admit of, the lengths of the sides calcu- lated by trigonometry, and their azimuths worked out. The work (when carried on on a very large scale) is still further complicated by the earth's surface being not a sphere but a spheroid. The accuracy of the triangulation istestedby what is called a"base of verification." Thatis a side of one of the small triangles is made to lie on suitaole ground, where it can be actually measured." Its length, as thus obtained, compared with that given bv calculation through the chain of triangles, shows what reliance can be placed on the intermediate work. ^ As instances: The triangulation commenced at the i^ough Foyle base in the North of Ireland was carried through a long Cham of triangles to a base of verification on Salisbury plain, and the actual measured length of the latter was found to differ only 5 inches from the length as calculated through several hundred miles of triangulation An original base was measured at Fire Island, near New York, and afterwards connected with a base of verifica- tion on Kent Island in Che^eake Bay. The actual distance between them was 2^08 miles, and the distance through the 32 intervening triangles 320. The difference between the computed and measured lengths of the base of verification was only 4 inches. In Algiers, two bases about ID kilometres long were connected by a chain of 88 triangles. Their calculated and measured distances agreed within 16 inches. If the country to be triangulated is very extensive- as, for instance, in the case of India-instead of covering It with a network of triangulation, it may>tersected in the first place by chains of triangles, either sfngle or double Base Lines. 123 and bases measured at certain places;usuallv where these chains meet. In India the chains run j^enerally either north and south or east and west, and form a great frame or lattice work on whicli to found the further survey of the country. A double chain of triangles forms, of course a series of quadrilateral figures, in each of which both the diagonals, as well as the sides, may be calculated. The following is a brief account of the measurements ot some celebrated base lines : In 1736 a base line had to be measured in Lapland for the purpose of finding the length of an arc of the meridian by tnangulation. A distance of about 9 miles was meas- ured m mid winter on the frozen surface of the River Tornea. By means of a standard toise brought from trance, a length of exactly 5 toises (about 32 feet) was marked on the inside wall of a hut, and eight rods of pine terminated with metal studs for contact, cut to this exact length. It had been previously ascertained that changes of temperature had no apparent effect on their length The surveying party was divided into two, each taking four rods, and two independent measurements of the base were made, the results agreeing within four inches The time occupied was seven days. The rods were probably placed end to end on the surface of the snow. The^me^j^ar^a base 7.6 miles long was measured near guito m Pertn at an altitude of nearly 8000 feet The work occupied 29 days. Rods 20 feet long, terminated at each end by copper plates for contact, were used. The rods were laid horizontally, changes of level being effected by a plummet suspended by a fine hair The rods were compared daily with a toise marked on an iron bar which had been laid off from a standard toise brought from Paris. This base was the commencement ot a chain of triangles for the measurement of a meri- dianal arc. Three years later another base, 6.4 miles long I ! 1 : I 124 Base Zines. itii was measured near the south end of this chnin .n^ , occup ed ten div^ Ti,« . »" tnis cnani, and only The trigonometrical survey of Grpot UrU • menced by the measurempn. f . '" '''^' ^^'^^ kngth of .he base when reduced .0 the TaW fand el! 40 links half :; Lh ;t : ™i :' 'r- ^°"?"""" °f chain was used as a standard of com°ariso„""T,^ ""5" was laid in five deal coffers carted on estl'es anV '" kep. stretched hya weigh, of aS pounds Th'e'ac, Z -eich^:iait^:xr-rs. '^'^- '"^- -' Carcassonne in the south. Four rods wer ied Th " The Lough Foyle base was measured with r.m > compensation bars; an arrangement in which th. ^ ". Base Lines. . 125 two platinum points at an invariable distance from each oti.er. ] he bars ^vcve avv.uv^cd in line on roller supports jn boxes ia.d o. trestles, and the intervals between the bars were measured by similar short compensating bars S.X mches lon-^ at each end of which was a microscope into the focus of which the platinum points of the measnr- n>R rods were bou^dit by means of micrometer screws. The hne commenced at a platinum dot let into a stone pillar, and the rods were kept in a true straight line by a -ransit theodolite. About 250 feet a day was measured on an average; 400 feet of the line was across a river the boxes bemg laid on piles about 5 feet apart. Ei^ht miles were measured thus, and two more were subse- quently added to the base by triangulation. The Salis- bury Plam base was measured in the same way. Colby's bars were subsequently used for ten bases in India, but were not found togive very reliable results there. An improvement on Colby's arrangement is the compen- satmg apparatus used in the United States coast survey It consists of a bar of brass and a bar of iron, a little less than SIX metres long and parallel to each other. The bars are joined together at one end. but free to move at the other Their cross-sections are so arranged that while they have equal absorbing surfaces their masses are in- versely as4^ specific heats, allowance being made for heir difference of conducting power. The brass bar is the lowest, and is carried on rollers mounted in suspend- ing stirups. The iron bar rests on small rollers fastened to It which run on the brass bar. M 126 Base Lines. =1 c W L~~±~- if' will be seen .„a. .hrt e/ ^c IS-o't" "'• " on the end of Z i baw/) ^tf? '"""/■""'="«-« w=re^.:e:-:,:x^^^ inner edR"atatn/a™r„°'"?'^',''"''='^e» W' ^'^ «> Thi= , ""'"f ^Sainst a contact iever (k) pivoted at (>). This lever, when pressed by the slidin,! i„V '" <='""'« with the short tail of the leveUkltuT" monnted on trunnions and not ba anced P '' ■" position of the sh-ding rod this bubble cotes^oth'e "?" and th,s position gives the true length oT he i °' bar. Another use of the level i. To ""^asunng pressure at the points of contae' p liT'r ^^T''' and level is attached tl,. " '• f ^"'^ e- To the lever indination of «« bar " """ "'"■^" ^'^^ ">e Base Lines. ,_ 127 f2\^ ^"' T '"''°''^ '" ^ spar-shaped^ublT^ tubular case the air-chamber between the two case s p e ventnjg rap.d changes of temperature. The end are closed, the ends only of the shdin, rods projecting T, level sector,and vernier, are read through glass doo The tubes are painted white and mounted on a pX of trestles. Two of these bars are used in measuring ' They are aligned by a transit. ^ ^ pa^atus"'thr'' ^^rr"''" '""^' "^^^"••'^^ -^^'^ ^^Is ap- paratus, the greatest supposable error was comiilted from re-^measurement, to be less than six-tenthsT an' •nch. On another base, six and three quarter miles Ion" the probable error was less than one-tenth of an "h' and the greatest sup.osable error less than three-tenths! This apparatus has been tested by measuring a base in Geo g,a three tm>es, twice in winter and once L summer at temperatures ranging from 18' to 107° Faht The discrepancies of the three measures with their re peclive means were, in millimetres,-8.xo,-o.32, and "^/x It has been found, however, that the apparatus is not quite perfect its true length depending on wl ethe "l^ t.-mperature is rising or falling. wneiner the The amount of accuracy to be aimed at in measuring a base depends on the extent of the survey. For sm H surveys It may be sufficient to measure the base two o witn a standard. The tapi; should be stretched earh .me to a constant tension by means of a spring balance It .s a good plan to mark the end of each chain on !,' small p,ece of plank, which is made to adhere ,o the gronnd by means of pointed spikes on its underLr?ace!„ nJ!l'V°i'' ""^'l ^'^"on^d. baked, boiled in drying oil pamted and varn.shed, may be used. They should eMer be levelled or have their angle of inclinatio'n read. I h" ^ //UL^fUt l^/^-c^^ r^. r e^ ":p)i=qi——|, nearly. If « be the number of rods in the base and n l=L- then the length of the base reduced to the sea level will be L li—±.AM.^ ('»), • , 1 r '~lt~}T~n~ "^^'"^ the mean height of all the rods. Then or Base Lines. 131 The base thus reduced is a curvp T« fi„^ lu 1 7 of its chord we should have tnubtrl, , "'^"' quantity. „amely,tlfe>1, dlvLet "ftiLTZhell^^ of the earth's radius. tne-flwaa^ u,,^ MEASUREMENT OF BASES BY SOUND. .7\'\ '^' t '°"^^ ""^^^^^ ^^^^h has sometimes to be adopted m hydrographic surveys of extensive shoa which have no pomts above water. It should, ifpossi^e onv be adopted m calm dry weather. The velocity o ' und m a:r ,s X089.42 feet per second at 32" Faht iW^'^^ effected by the wind, the barometer pressu e a7?tt hygromefc condition of the air. The ob e'rve^s Ire posted at both ends of the base and are prov ded wUh fnd"' TfT 'I' *h^™°-^t-- When the gun at one end IS fired the observer at the other notes thelnterval in seconds and fractions between the flash and tie reptt The guns are fired alternately from both ends a Teas; three times, a preparatory signal being given. Ject Jd t thft' "''"!' °' ^°""' ^^^^" ^^-^ --t be quandt^ temperature (O by multiplying it by the *^i+(^°— 32°) X0.00208. nhv/h°T *^^' ^^'*/"'" ^' '^^ ^°"^^ted velocity multi- plied by the mean of the observed intervals of time The errors of observation are always considerable, but are no greater for long distances than for short ones. ASTRONOMICAL BASE LINES. In cases where no suitable ground for a measured base s available two convenient stations may be sdected as the ends of an imaginary base line, and their Ta^tude and longitude, with the azimuth of one from the other ascertained by astronomical observations. We ha ,' then have the length and position of the base with more or less accuracy, and a triangulation can be carried on m \\ i\ li^r !!( 132 Base Lines. from It. The base chosen should be as long as possible but not greater than one degree. None of the sides of the triangles should be greater than the base. The azi- muths of the sides being known, the positions of the observed points can be plotted by co-ordinates. If the zenith telescope and portable transit telescope are used the latitude can be determined within lo", the longitude and azimuth within 30". With the sextant these errors are at least doubled. Differences of longitude may be determined by flashing signals. CHAPTER III. ii TRI ANGULATION. Having discus • • ,',e measurement of base lines we ^thT./° " u'" "^' triangulation. It is evident that the latter may be commenced without waiting to com- plete the former. The first thing to be done is'o select he stations and to erect the necessary points to be ob- served, or "signals" as they are called. In a hilly country the mountam tops naturally offer the best stations, as being conspicuous objects and affording the most distant views In this case the sue of the triangles is only limited by the distance at which the signals can be observed. Thus, n the Ordnance Survey of Ireland the average length of the sides of the primary triangles was 60 miles, while some were ^, h as 100. In the triangulation which was carried in 1879 across the Mediterranean between Spain oblrT': ':.•'"'""' °^ '^' ^^^^*^'^ "^ht, signals were observed at a distance of 170 miles. In a flat country lofty signals have to be erected, not only that they may be mutually visible.' but in order that he eart'h Jffh* "'^^ ""^ll^^l *°° ^^°^^ *« ^he surface of the earth, as they would be thereby too much affected by refrac ,on. In the U. S. Coast Survey six feet is consid- ered the limit advisable. If K, K', are the heights of two III ;. I :i n 134 Triangulation. signals m feet and d their distance in miles, then on a flat country cr ov. water, they will not. unier ordCarv H Vh + 1/ . ) The most difficult country of all in which Formerly, conspicuous objects, such as the points of years th.s has not been done, because in all large triangles !t IS necessary to measure all the three angles and tl k cann t ,,^,1 be done directly in the case ofTuch object^ The form of the signals varies much. Whatever Idid 1,; used the centre of the theodolite must be placed eiacdy under or oyer the centre of the station, and if a scaffold ing has to be employed the portion on which theTn!^!' men .s supported must be disconnected with that on wh.ch the observers stand. One kind of signal s a ver- t cal pole with tr.pod supports, the pole being set up wi h Its sum,mt exactly over the station. It may bHur- mounted b^ two circular disks of iron at right anls to each otherf A piece of square boarding, paintedthite with a vertical black stripe about four inches wide. cL be seen a long way off. Flags may be used, but are not al- ways easy to see. A good form of signal is a hemisphere of silvered copper with its axis vertical. This will reflect the rays of the sun in whatever position the latter may b bu a c. rrection for "phase" will be required, as the rays will be reflected from different parts of the hemisphere according to the time of day. The ordinary signa used m the United States is a pole lo to 25 feet hTeh Z mounted by a flag, and steadied by bracel Witit^'c; to Its diameter, the rul. is that for triangles with shies not exceeding five rniles it shouJd not be more than five mches. f more than five miles, five to eight inches Various other forms of special signals are usedfn the US Coast Survey. Amongst others may be mentioned a ^^^/. O-i rt <^i) -^^'f-l^ ^^j-i-r a^ r*i^< y- .^ ^. ^<_e Triangulalion. 135 pyramid of four poles, with its upper portion ^a^^^^^^ doHte r nl 'h '" ^P""^' '''^'''y ""^-- -hich the th :: dohte ,s placed. In England double scaffoldings as high as 80 feet were used, the inner scaffolding c.rrvin^ the instrument and the outer one the ob eLrs Jn cov d with im^penetrableT::ts: atd^ scIffTdU^:^^ much as 146 feet high had to be erected On thf o .he Western States towers havell t'o ..ZZTZ has been done ,n India, where solid towers were used at first, but wereafterwards superseded byhollow ones which stZ '"^'"^'---'^.'o.b^ centred' verti:;ro;rr era v^^'h- ♦ . K '' °' '"S^o^ctrical stations are gen! erally indicated by a well-defined mark on the upper sur face of a block of stone buried at a sufficient di tTnce b"; wer ma'kt^bv fl".^^"^"■^"♦™"'''"'-'°"">es::Ho„s were marked by flat-topped cones of masonry havin- n vert^al a.al aperture communicating with [he station diJl'""!!*"' '""'™' ""^y ^^ ""'^^""i "siWe at a great .tis:.t r :nife^» r. rf "■{^'" across the Mediterranean Srldy JliZ ^'^^^ torW„r' " rf "" ^"^^"■^Si-'es of six-hnrse powe working magneto-electric machines. These hVhfc ctstrr^^^^ '-'-'- - inch:ri:ta;rt::: conTe urf: e si vTeT' Th"'^ '^" °' t^^ ^''' ^^^ corrected th. 1 ,^^\ ^^^ curvatures of the surfaces corrected the lens for spherical aberration, and it threw out a cone of white light, having an ampHtude of ! whach was directed on the distant station by a teles ot' A refractmg lens, eight inches in diameter was T' -ec^ and threw the light one hundred al^^nns There were two Spanish stations fiftv n.iles Ipar " ^Mulhacen. xx,42o feet high, and Tetica. 6,8.0 feet. The *.■ 1 I' 136 TrtangulatioH. two Algerian stations, 3,730 and 1,920 feet, were 66 miles apart, and were each distant from Mulhacen about ^70 miles. The labour of ti .nsporting the necessary machin- ery, wood, water, &c., to such a height as Mulhacen was very great It was twenty days after everything had been got ready before the first signal light was made out across the sea. After that the observations were carried on un- interruptedly. In France, night observations have been carried on by means of a petroleum lamp placed in the focus of a refracting lens of eight inches diameter. MEASURING THE ANGLES. Oflate years the only instruments used for measuring the angles of a triangulation have been theodolites of various sizes ; the larger natures being really "alt-azimuth" instruments. The more important and extend -1 the sur- vey the larger and more delicate are the instroa.ents em- p oyed. In the great triangulation of India theodolites of 18 and 36 inches diameter were used, the average length of the triangle sides being about 30 miles. For the bpanish-AIgienan triangulation they had theodolites of 16 inches diameter read by four micrometers. In the United States Coast Survey the large theodolites have diameters of 24 and 30 inches. For the secondary and tertiary triangles smaller instruments are used The method of taking the angles varies with the nature of the instrument The smaller ones have usually two verniers. Ihose of about 8 inches diameter have three, while the arcs of the larger ones are read by micrometers, of which some have as many as five. In all cases errors due to unequal graduation and false centreing are almost entire- ly eliminated by the practice of reading all the verniers or micrometers, and taking the same angles from differ- ent parts of the arc. It is usual to measure all important angles a large number of times, ^^7 a^^/A,,^,^^^ "/C^X^ /- Triangulation. — — — — __i?! Of the smaller theodolites there are two kinds th. r. required In a reiterating theodolite the lower nlate i, fixed to the stand, and when the instrumentTs set un for he purpose of measuring a horizontal angL it s "ite a chance what point of the graduation the angle willtave to be measured frnm tj,;^ t • '^"S'c win nave the oth.r f ^' ^ ^°™ '^ "''^ so convenient as the other for general purposes, but it has the advantaire of used on he r ^^^.^■^"^h reiterating transit theodolite! verniers * ^f ^n 'T" • ''°'"""^"* '""''^^^ ^^^^ three sconrthen t T '^ '"'^ °^' °" ^^^h' ^^^ the tele- Shan h. '"''^ °"'' ""'^ '^^ ^"^^^ remeasured, we shall have six measures from six equidistant parts of Z arc the mean of which is taken. In all cases .ft.r reading an angle or a round of angles he el 'non should be again directed on the first obg^btvd:.'' LTh^Xer ^^^ '-'^^ - ''' — -^ ^^" -h- The method of "repeating" an angle is this. When the telescope has been directed on the second or H^M hand, object, and clamped, instead of reading the vemfet the lower plate is set free, and the two togetLr rlXed till the telescope is again set on the left hand object In,r:?^"'' '' *^'" ^'""^P^^' th« "PPer one settee and he telescope directed on the right hand obiect Thl reading-or the repeatmg process may be continued for divided hT."' '™"' ^"^ *^^ "^°^^ -- passed ovr divided by the number of repetitions. The object of thT. process IS to diminate errors of observation aTd ,radt tion. Owing, probably, to slipping of the plates il not usually give such good resul Is m!ghtt'^:;p.etd: t 1 138 Trtangulation. TO REDUCE A MEASURED ANGLE TO THE CENTRE OF A STATION. It may happen than an inaccessible object— such as the summit of a church spire— has to be used as an angle of an important triangle. It cannot, of course, be meas- ured directly, but it may be found indirectly as follows : Let ABC be the triangle and A the inaccessible point. Take a contiguous point A' and measure the angles ABC, BCA, BA'C, AA'B. Calculate or otherwise obtain the distance AA' on plan. CallBAC, A; BA'C, A '; ABA', a; and ACA', /3. Now A + a=A'-f/3. Therefore A=A'+/9— a. Also, AB and AC are known, and f AB sin a=AA' sin AA'B (AC sin /9=AA' sin AA'C pig. 33. or, since a and /9 are very small angles, if they are taken in seconds, . AB X a sin I'l =AA' sin AA'B ACx/9sini =AA' sin AA'C Therefore. A=A'-^^' "" ^^^^+ AA' sin AA 'C AB sin 1" ^ AC sin i" " CORRECTION FOR PHASE OI SIGNAL. If the sun shines on a reflecting signal— such as a poHshed cone, cylinder, or sphere— the point observed will, in general, be on one side of the true signal, and a correc- tion will have to be made in the measured angle. The following is the rule in the case of a cylinder. Let r be the radius of the base of the cylinder, Z the horizontal angle at the point of observation between the sun and the signal, and D the distance. >i Triangulation. 139 r cos— • 2 Then, the correction = ± D sin i" The proof is very simple. ^Injhe case of a hemisphere the value of r will depend If we call the latter A, r will on the sun's altitude. become r cos ^ . which must be substituted for r in the above equation. TC REDUCE AN INCLINED ANGLE TO THE HORIZONTAL PLANE. It often happens, as in the case of angles meas- ured with the sextant or repeatinn^ circle, that the observed an^^le is inclined to the horizontal, and a reduction is necessary to get the true horizontal angle. In Fig. 34, let O be the observer's position, a and 6the objects, anda Oft ^'^•34. the observed angle. If Z is the zenith, and vertical arcs are drawn tough a and b, meeting the horizon in A and B, then A^^is the angle required, aZb is a. spherical triangle, and by measuring the vertical angles Aa, Bb we shall have its three sides, since ZA and Z B are each 90°. Also, aZb=AO B. If we call ab, h ; Za, z; and Zb, z , we can obtain aZ b from the equation (I Z_b_ fsinjs— r)_sinj(s-_£)| J I sin s sin z j '■f sm where s = ^±^±i' 2 The arcs Aa, Bb are generally small, and the differ- $ 140 Triangulaiion. ence of z and / therefore also small. The arcs may therefore be substituted for the sines, and we have for the correction (in seconds) AOB-;, = (9o--!±f:} '.anisin r"-(-:^] "co.^ sin ," This formula is applicable when z and .-' are witWn ^' of go . -^ If one of the objects is on the horizon we shall have AOB— A 2|45° cot /{ sin i" J{, in addition, the angle h is 90° the correction will be THE SPHERICAL EXCESS. The angles of a triangle measured by the theodolite are those of a spherical triangle ; the reason being that at each station the horizontal plate when levelled is tangen- tial to the earth's surface at that point. We must therefore expect to find that the three angles of a large triangle, when added together, amount to more than 180- and this IS actually tne case. The difference is called the spherical excess." From spherical trigonometry we know that Its amount is directly proportional to the area of the triangle. In small triangles it is inappreciable. An equilateral triangle of 13 miles a side would have an excess of only one second. I or one of 102 miles it would be one minute. Taking for granted that the spherical excesses of two triangles are as their areas we can easily find the excess for a triangle of area s-thus : A trirectangular triangle has a surface of one-eighth that of the sphere, or ^',and its excess is 90', or 324000". The excess, in seconds, will therefore be equal to ^^Jll^x .; . and . being, of course, in the same unit of measure. g Triangulation. Since 141 obtained with suffiZt a^X;' J'Th" '' "^^ '' treating the tHan„e as it rr/pitetne^"^^^^ thus use either of the formul* ""^^^ ._ a b sin C 5 ■■■ " ~' 2 «2 sin B sin C or 5 — 2 sin (B-f cr according to the data given X Ifj-K^jU, o •^-'i ^»- ^Itx. C--.^lj ^^ Q^ *■ 2 ' ■ r2 sin 1^ become account. The expression will then «^sin CJi-(-^2.co^_L) 2 y2 sin 1' CORRECTING THE ANGLES OF A TRIANGLE. In practice the sum of the three measured angles of a tnang e IS never what it ought to be, and they have to 1 e corrected. Supposing that all three angles have bel measured with equal care, the plan adopted is to add o or subtract from each of the angles one third J 1 excess or deficiency. Thus, if E" ^ariie c ,cu a ted p encal excess the three angles ought to amounrto HaL' ^"PP^^^"^ that they amounted to i8o»-f„« andthatn weregreaterthan E. Then we should subtract from each angle -^— ^ 3 If some angles have been measured oltener, or with ^greater care, than others, the amount of correction to be /^^-^-^^ /V>^/^//^ >i^fc^^^) ^^^ 142 Triangulation. 11 applied to each will be to the results of the inverse measurements ly as the weights attached In tne Spain-Algiers quadrilateral triangulation the spherical excesses of the four triangles were 43".5o;6o".7;7o".73;54".i6 and the errors of the sums of the observed angles were +0.18 -o".54 -f.1-.84 +i".i2 CALCULATING THE SIDES OF THE TRIANGLES. The next Step is the calculation of the sides of the triangles Treating the latter as spherical this may be done m three ways. • 1. Using the ordinary formulas of spherical trigonome- try. This is a very laborious method, and others which are simpler give equally good results. 2. Delambre's method. This consists i„ taking the chords of the sides, calculating the angles they make with each other, and solving the plane triangle thus found. To reduce an arc a to its chord we have Choru=2 sin ^ a or, if the arc be in terms of the radius Chord =a — -^ a^ . The angles made by the chords are obtained by a well- known problem in spherical trigonometry. 3rd method, by Legendre's Theorem ; which is, that in any spherical triangle, the sides of which are ve y small compared to the radius of the sphere, if each of the angles be diminished by one-third of the spherical excess, he sines of these angles will be proportional to the engths of the opposite sides; and the triangle may therefore be calculated as if it were a plane one. .u ^l *-^'l^ T?'''^^ "^^'^ "'^^ •" *he French surveys. In the British Ordnance survey the triangles were generally a TC 2^/'*-a^77t^'-sl- Triangulation. j.. calc^ulated by the second meU^d and cl^c'k^TV^e Legendie's theorem gives very nearly accurate results. In a triangle of which the sides were 220, i«o. and 60 miles, the errors in the two long sides, as calculated by this method from the short side. >vouId be only three- tenths of a foot. The following investigation shows i^i dcr vhat circum- stances small errors in the measureirents of tV- andes of a triangle have the least effect . pc the calculated lengths of the sides. Suppose that in a triangle a b c we ha- e the side b as a measured base, and measure the angles A and C; we have a sin B=6 sin A If we suppose b to have been correctly measured we may treat it as a constant ; and under this supposition It we differentiate the above equation we shall "et or, since - a sin B sin A da=a cot A d A— a cot B at'The latitude L' is not quite the true latitude of A'; for the latter is A' N' Ol or the angle made by the normal A' N' with N' Q', while the latitude given by equation («) is the angle A' N Q The correction of the latitude (j^) is the angle N A' N'- for A'Ng-A'N'Q'=A'RQ'-A'N'Q'=N'A'R ^ "^ "^ ' ^^"^ andsinc^=^^^"J'JilA' -vT ./..,/.. ^ N' A' B^fo»vestigati«g the exact value of this angle it should be noted that when the geodesic line K is more m }M l> ' I 4-'< 5' 148 Geodetic Latitudes, &c. than half a degree its amplitude in latitude on the sphere —say ^L— becomes a different quantity— say A L— on the ellipsoid, and that these two amplitudes of arcs of the same length being inversely proportional to their radii of curvature N, R, we have A L : (;L::N : R;:i : — ^~^ — , , 1— eS sin2 L whence we have, very nearly A L-i L (i+..yiC€^<^,j^^ _ ^KVi^A. >*¥ Za^ / /^ ■ wC'-Ta y^'^ zV'-^ 'r / ' /<' uj./ ^a,/ Av- / ' ^« •i^* ' sphere l^L — on arcs of sir radii :os»L) ■ when to de- educe Now V ) / >-'fi¥IHA^^4*, le as Mi . Geodetic Latitudes, S-c t..« ~ . , ^ *49 The imaginary sphere used in the abo^e in^i^tio^n wil,o course, coincide with the spheroid for the parallel of latitude through the point A. Any plane pass ng through the normal will cut the surface of the sphere in the arc of a great circle, and the spheroid in aline, which for^about three degrees, will be practically a geodesic The following is another way of treating the sub- ject Instead of taking the normal at one of the po nts A A as the radius of the imaginary sphere let us take th normal at the point B, mid-way between them L n F^ 36, and for the sake ^^— — """rig. of simplicity let these points be on the same meridian. Let A N, A' N' be the normals at A A', produce them to Z and Z' respcc tively, and draw A c, A' e parallel to the f»aj or ax i s O E. The astronomical lati" tudesof the two points are Z Ae,Z' A' e. If now we draw B C the normal at B C wfll fall between N and N'. The curve g ve„ ,„ the figure is the elliptical meridian. The circuh we n. u , .:':a:snhXrYor;::..r;- mag,nary sphere, one being less and the other grea^^ «.an the lamudes on the spheroid. The dikC ^^ -^1^/ ' ""^r^^'-d 'he same. Let each be it' already investigated. In what next follows K is the distance A A' in yards o any tvvo stations A, A', u the same distance in seconds ^^^' ^ 'he radius of curvature of the meridian, N the normal (both in yards), . the eccentricity (=0.0817), and a the equatorial radius. Equation («') gives us the values of « and L', (b') eives us M and (0 gives Z'. If we neglect the de'non in' o of the fraction m (c) we have Z'=i8o' + Z—{M'~M) sin ^L + L') orZ'«.8o".Z-^-ji^^sini(L.L') of Jh^.' '^'' '"r °^ 1^'' "^""*'°"' ^^^^h '« the difference of the azimuths at the two stations, is the convergence of their meridians. ^ ^ ^^. .x^ .^ ^ ♦-J"- P»«.*«_, !r ^^ e--«-««1^ .^y^^' -<^^ ^//l/Lyi^y \ y. *) N sin I" X Z'=i3o°+Z± ._ ^ N sin i" The sphere described with radius equal to the normal or the mean latitude of two stations,\ a^d B maTbe used m the next three problems. U.^ 7W X. kSl^"^ ^^ I. GIVEN THE LATITUDES AND LONGITUDES OF TWn POINTS TO FIND THE LENGTH AND DIRECTinv nS THE LINE JOINING THEM. DIRECTION OF Here we have given L. L', M. and M', and from Land L' we obtain 5. ^ We have then to find / and /' from the equations /-L-l-and/'-L'+i- Let x" be the number of seconds in the arc np«c,-n^ through the point of which L' is the latitude andperp n d.cular to the meridian through the other point ^^ Let y be the number of seconds in the portion of this meridian between L and the foot of this peVdicuIar Let a;, y, be the same quantities in linear units N-th^ A^^ A the nTxtl^fjfget hatbe^^li^ Fractions ofJJnds ha^| Sel^ftte t'lJ'^ ^-^ ^"^^ying. ^ *-^ yt^»t. 't.-t-^fcr,* ;*&«. C-cn.'T.e^e, >^^ ^/«^ ^^ .^Cr .^^^^^ <5k ^^--^^i^^ -*cL-^ ^^^t^^^ :?/^-x^'S=^ l£j^ V ^^•^ ■ : I i-'.i 152 Geodetic Latitudes, &c. Then we shall have .r"— (M'— M)cos/' y'-^l—l'—l sin i" X2' tan / X'^x" N sin I" y—y" N sin i" tan Z^^ u"= x" yll sin 2 cos~Z K=>=w' N sin I" EXAMPLE. Given L=49' 4' 25" L =49 22 33 M~M, or difference of longitude-38' 47«=2^27- to find Z and K ^ ' Here L + L'-^gS" 26' 58" HL+L)=49 1329 L — L». o 18 8 HW— L)=o 9 4=544« To find the value of— log 62—7.81085 , log \ (L'-L)=2.73549 2logcosJ(L + L')=9.62994 Jog Y-^o- 17628 -^=-1 -5 Z=T ^ ^^ 7- =49° 4' 26".5 (5 being negative) /'=L> ^=4q"22'3l".5 Geodetic Latitudes, &c. To find x" Log (M'—M) =3.3668785 log cos /'=9.8i3647o log ^=3-i8o7^ X =1515" To find the value of the 2nd term of/ log i Sim "=4.38454 2 log* "=6.36105 log tan /=o.o6i97 log 2nd term=o.8o756 2nd term="o° o' 6" I'—l-^o 18 5 y''=0 18 II=logi- To find the azimuth Z Log a;"=3. 1805255 ]ogy'=3.o378887 log -y =0.1426368 Z=i25°45'2i" To find log N sin i" Log N (in yards)=6.8443224 log sin i"=4.6855749 Log N sin i"= 1.5298973 To find log u" Logj;"=3.o378887 log cos 2=9.7666596 T a J ^J°S «*''=3-27I229I To find K log «"=3.27I229I log N sm i"=i.5298973 4.8c" 64 iiC= 632.^.6 yards. 153 li ii 'i I- 1 1 nil 154 Geodetic Latitude'^, f^c. To find the co-ordimtcs. Value of X. Log .V '=3.1805255 logNsmi =1.5298973 log Jf=47i04228 ^=51336 yards Value of y. Log>^;;=3.o378887 log N sin I =1.5298973 log v=4. 6^77860 J -" jO^cj yards TO COMPUTE THE DISTANCE BETWEKN TWO POINTS KNOWING THEIR LATITUDES AND THE AZIMUTH OF ONE FROM THE OTHER. Let L and L' be the latitudes, Z the azimuth, and let 2 Then we shall have, as before — ==^iikzLlcosy a 2 2 _(i— e» sin'T)^ /'=L'+- Assume tan I sin /' then, sin (^— «"^.!l£i_sin w .... "in / '^ which gives m; and K--w" N 1 i'' The algebraic sign of cos Z will determine the sign of tP and consequently, whether u" h .0 be added fo or «ub.' tracted from ~ '^' 5 "=1085" L + L"^98 2655 log«8_7.8io85 log(L-L")=3.o353? 2logcosi(L+L'/)=9.629g4 log ^=0.47610 L'—L"— ^=49° 22 '33" '=L~^=49° 4' 26 ".5 ~ +~^ =49 22 31 .5 To find u" and K— log w=3.3668785 log cos/-=9.8i3647i co-log sin Z-=-o.o907036 log m"=3.27I2292 log N sin I "=-1.5298973 log K=4.8oii265 K=63226 yards. 1 _ Geodetic Latitudes, 6-c. 1*7 On the North American boundary survey in 1845 the foIow,n,^ method was employed to find the mutual azi- muths of two distant points the latitudes and longitudes of which were known. Let A and B be the two points, of which B is the northern, and P the pole. Then, treating the earth a a fdrpTl r'/r '^'T^ tnangleSABthe two s.dcs P A, I B, and the angle A P J3 given, and have to hnd the angles A. B. This is donelby the usual forull^ tan i (A+B)^ ^ AP-BP cos — .^ _ 2 p AP-t-BP^ ^°^ 2 cos * 2 tan ^ (B-A)^ sin sm AP-BP A P AP+BP ^ ^°' 2~ which give -^ and 2 B— A 2 Then,A=^+-^_BrA 2 2 B=:A±^, B-A 2 "^ 2 To correct these azimuths for the earth's spheroidal form take 9o»- A and B-go", and calculate th. angles «, /J, from the formulas sm a= sin AP sin j3= sin BP ^75 -■'- i/7J 'l^hen. if A' and B' are the true spheroidal azimuths, tan (90 — A')=-cos « tan (90*- A) tan (B'-9o=)=cos ,3 tan (B-90") This method is very useful when a long line has to be cut from one pomt to another through forests. M II :< 1 158 Geodetic Latitudes, &c. To find the accurate length of the arc on the surface of the earth between two very distant points of known lati- tude and longitude is a very difficult and not very useful problem. It is, however, often advisable to calculate the distances between stations that are within the limits of tnangulation, as a check upon the geodesical operations • and in the case of an extended line of coast, or in a wild and difficult country where triangulation is impossible, this problem is most useful for the purpose of laying down upon paper a number of fixed points from which to carry on a survey. "^ In the triangle PAB mentioned in the last article we have, as before, the sides PA, PB, and the angle P as data. By solving the triangle we obtain the length of the arc AB. If the azimuths can be observed at the two stations the accuracy of the result will be greatly increas ed, and we can obtain the difference of longitude of the two stations as follows :-It may be proved that the sphe rical excess in a spheroidal triangle is equal to that in a spherical triangle whose vertices have the same astrono micai latitude and the same differences of longitude • from whence results the rule PA-PB A+B tan — = 2 cos- cos PA + PB X cot cos i diff. lat -- xcot A-hB sin ^ sum of lat. 2 which gives P, or the difference of longitude. As a rule, a small error in the latitudes is of no import- ance unless the latitudes are small : but the azimuths must be observed with the greatest accuracy. The angle P being knov/n we can get the length of the arc AB, and must then convert it into distance on the earth's surface using the radius of curvature of the arc for the mean lati- tude. n^-^iir^l'Kf.nr-J^'m-R-v: Devillch Methods. 159 It may be observed that when dealing with the sphere we have the definite equations of spherical trigonometry to work with ; while, when a spheroidal surface is in question, we have only approximate formula. In most of the equations employed in higher geodesy the right hand side consists of the first few terms of a converging series, the remainder being so small that they may be omitted without causing anv appreciable error. Thus in calculating the length of a geodesic line between two dis- tant points the smaller terms of the series might amount to only a few inches in 100 miles. Captain Deville in his "Examples of Astronomic and Geodetic Calculations" gives some very simple methods of solving certain problems in Geodesy by means of tables of logarithms of the convergence of meridians for onE chain departure, and tables of the value of a chain in seconds of latitude and in seconds of longitude at different latitudes. By departure is meant, of course the distance one point is east or west of the other If a great circle (not being a meridian or the equator) is drawn on the earth's surface it will cut each meridian it crosses at a different angle according to the laitude of the point of section: in other words its azirmth is continually changit.g: and if we take two points, A and B, on this great cir.ie, and P is the pole, the convergence between A and B will be, practically, 180 -(PAB + PBA.) If the two points are in the same latitude, and one chain dis- tant from each other, each of the angles A and B will be less than 90° by half the convergence. If the distance be constant the convergence will increase as we recede from the equator (where it is nothing) towards the poles. In problems involving two stations of different latitude the convergence used it that for the mean of the two latitudes. The examples given by Captain Deville are worked out by logarithms. In the following selection of problems ' the principle only of the method is indicated M i6o Deville's Methods. Prob. I.— To find the convergence of meridians between two points of given latitude. Here we have only to find by a traverse table the de- parture in chains and multiply it by the convergence for one cham for the mean latitude. Prob. 2.— To refer to the meridian of a point B an azimuth reckoned from the meridian of another point A. Calculate the convergence between the two points and add or subtract it from the given azimuth according as B IS east or west of A. Prob. 3.-Given the latitude and longitude of a station A, and the azimuth and distance of another station B to find the latitude and longitude of the latter. The distance and azimuth being given we can find the departure and distance in latitude of B approximately by the traverse table, and have the approximate mean lati- tude We next find the mean azimuth by multiplying the departure by the convergence for one chain at the mean latitude, and applying the convergence thus ob- amed to the azimuth of B at A, which gives the azimuth ot A at B, and hence the mean azimuth. To get the correct latitude of B we multiply the dis- tance by the cosine of the mean azimuth and by the value of one chain in seconds of latitude. This gives the differ- erence of latitude of the two stations in seconds. Similarly, the difference of longitude of the stations is found by multiplying the distance by the sine of the mean azimuth and by the value of one chain in seconds of longitude. Prob. 4.~To correct a traverse by the sun's azimuth. On a traverse survey of any extent the direction of the lines must be corrected from time to time by astronomi- cal observations, usually either of the sun or the pole star Deville's Methods. . ^_ i6i If the traverse is commenced at a station A with a kn^ orientation, and at another station B an observation is hnnM!?'^''r"'"'\°'"^'""'"^ °^*^^ ^^"^ ^hus obtained should differ from the azimuth as carried on through the raverse from A by the convergence of meridians between the stations Should there prove to be any error it should be equally distributed among the courses by dividing it by the number of stations. Multiplying the result by the number of any course gives the correction for that course As an example :-A traverse of seven courses was made in a westerly direction. At station 8, or the end of the 7th course, the sun was observed, and its azimuth found to be ^7^ II 50 , the horizontal plate reading being 267* =;q' to" Required the error of orientation. '^ / oy • On calculating the convergence between stations i and 8 It was found to be 49 ' 5". If the traverse had been cor- rectly run the sun's azimuth plus the convergence would nave been thp camp oc fii^ ^v.i-^ r . 267* 11' 50" 49 5 268 267 055 5910 MS have been the same as the plate reading; but the latter was i' 45" too little. One seventh of this, or 15". is the correction for each course, and we have to add 15" to the plate reading of the first course, 30" to that of the second, and so on. Prob. 5.— When running a line to correct its direction by the sun's azimuth. Unless the line is a north and south one its azimuth will be continually changing from point to point Its direction can be checked at any time by finding its azi- muth astronomically to ascertain if this is what it ought to be after allowing for the convergence. The first step is to find the approximate difference of latitude from the distance ch^. n-^d and the azimuth at which the line started. Tlv will give the latitude of the station and I 1 62 Devtlle's Methods. the mean latitude approximately. The latter being known, the azimuth and distance give the convergence, which being applied to the initial azimuth the true azi- muth is obtained. Prob. 6.— To lay out a given figure on the ground, cor- recting the courses by astronomical observations. Take as an instance a square ABCD, the side AB being commenced at A with a given azimuth. The course is to be corrected by observations at the other three corners. The convergence between A and B being found in the usual manner and applied to the original azimuth (in addition to the angle at he corner) gives us the tfzimuth of BC. Similarly, the convergence between A and C will give us the azimuth of CD ; and so on. Prob. 7.— To lay out a parallel of latitude by chords of a given length. The angle of deflection between two chords is the con- vergence of meridians for the length of a chord, and the azimuths will be 90° minus half the convergence and 270* plus half the convergence. The convergence is found in the usual way. Prob. 8.— To lay out a parallel of latitude by offsets. A parallel may be laid out by running a line perpendicu- lar to a meridian and measuring offsets towards the near- est pole. The length of an offset is its distance from the meridian multiplied by the sine of half the convergence for that distance; or (since the distance is in this case the same as the departure) the square of the distance multi- plied by the sine of half the convergence for one chain. As this angle is small the logaritiim of its sine is ob- tained by adding the logarithm of the sine of half a second to the logarithm of the convergence for one chain de- parture. When the offsets are equidistant any one of them may Deville's Methods. 163 be obtained by multiplying the first one by the sq^^IiT^ the nnmber of the offset. It is almost superfluous to point out that in practice all these problems are worked out by means of logarithms. TO FIND THE AREA OF A PORTION OF THE SURFACE OF A SPHERE BOUNDED BY TWO PARALLELS OF LATI^ TUDES AND TWO MERIDIANS (SPHERICAL SOLUTION.) Let AB and CD be the meridians and AC BD the parallels. Let f be the latitude of A, f of B and n' the difference of longitude of the meridians. Now the area of the whole portion of the surface com- prised between two parallels is equal to the area of the por- tion of the circum- scribing cylinder (the axis of which is the polar axis) contained between the planes of the parallels produced to meet it. {Vide second figure showing a section, in which a is the point A and b the point B.) Let r be the radius of the sphere and h the perpendicu- lar distance between the planes. Then the area of the spherical zone will be zn rxh =2 71 rxr (sin ip'—sin ip) ="2 -T r2 (sin ^'— sin f) :, the area of the portion between the two meridians will be -^^-(sin f—ainip) ' i 'I m i6^ Offsets to a Parallel. TO FIND THE OFFSETS TO A PARALLEL OF LATITUDE. Let PA, PBC, be meridians, AB a portion of the parallel, AC a por- tion of a great circle touching the parallel at A. It is required at a given latitude to find the offset BC for a given dis- tance AC. i'ig. 38-. Let X be the circular measure of AC y do. do. BC ^ do. do. PA AC and BC are very small. In triangle PCA we have cos PC--cos / cos x =cos / (i~-- ) nearly. Therefore cos I— cos PC=cos l~ or.3i„-i±rc3i„^BC 2 2 2 -cos / X' or 2 sin /-^^'^cos/, nearly. 2 "" 2 therefore, y=^ x^ cot / (or, if ^ and 3/ are measured lengths, and R is the radius of the earth, jy=^- cot/) Next join AB by a great circle arc. The angle BAC will be half the convergence, and AB=AC, approximately. Draw PD bisectmg P, and therefore at right angles to AB. In the triangle APD we have D— 90* convergence 2 and rAD= ■90"- U Offsetts to a Parallel. Therefore, cos PAD=tan AD cot /, 165 convergence sm - ^ _AD cot /, approximately =i X cot / Therefore, y^^ x^- cot /-.v sin ^o"J_?'"?ei^ 2 This is equally true if .v and j- are measured lengths. I ; ' 4 ^ CHAPTER V. I METHODS OF DELINEATING A SPHERICAL SURFACE ON A PLANE. Since the surface of the globe is spherical, and as the surface of a sphere cannot be rolled out flat, like that of a cone, it is evident that maps of any large tract of coun- try drawn on a flat sheet of paper cannot be made to ex- actly represent the relative position of the various points. It is necessary, therefore, to resort to some device in order that thejpoints on the map may have as nearly as possible the same relative position to each other as the corres- ponding points on the earth's surface. One method is to represent the points and lines of the sphere according to the rules of perspective, or as they would appear to the eye at some particular position with reference to the sphere and the plane of projection. Such a method is called q. projection. The principal pro- jections of the sphere are the "orthographic," "stereo- graphic," "central or gnomonic" and "globular." A second method is to lay down the points on the map according to some assumed mathematical law, the con- dition to be fulfilled being that the parts of the spherical surface to be represented, and their representations on the map, shall be similar in their small elements. To this \ a F d o tl SI tl e( ol ci th Projections. 167 class belongs M..c«^o.'. Projection, in which the meridians are represented by equi-distant parallel straight lines, and the parallels of latitude by parallel straight lines at ight angles to the meridians, but of which the distances from each other mcrease in going north or south from the equator m such a proportion as always to give the true beanngs of places from one another. The third method is to suppose a portion of the earth's surface to be a portion of the surface of a cone whose ax,s coincides with that of the earth, and whose vertex is somewhere beyond the pole, while its surface cuts or ouches the sphere at certain points. The conical sur- face is then supposed to be developed as a plane, which Ts thV^. r ^^T'"'^ '" '^^'^ P^^«^ ^« ^h- one known as the "ordinary polyconic." inlltn^''^'^?^'"' ^''^''''^■'" ^' ^^P^y the one employed m plans and elevations. When used for the delineation finL H ';"' "'^'r '''' ''' '' ^"PP^^^d to be at an in finue distance, so that the rays of light are parallel, the plane of projection being perpendicular to their direc ion ler^rofth^''^^^'^Pl^"-^P-i-t- is usuX* either that of the equator or of a meridian. When a hemi- sphere IS projected on its base in this manner the relative positions of points near the centre are given wi'h o 7 able accuracy, but those near the circumference are col" Pletely distorted. The laws of this projection are easUy deduced. Amongst others it is evident that in the ca e of a hemisphere projected on its base all circles nas! through the pole of the hemisphere are pro^'ted "^ straight lines intersecting at the centre. cLles having q:altcleV^ll!^l° ^'" 1 ''' '-^ -'^ P-^-'eTf equal circles. Ail other circles are oroierfpH ce .11 • ft i68 Projections. U4 i Stereographic Projection, -In this projection the eye is supposed to be situated at the surface of the sphere, and the plane of the projection is that of the j^reat circle which IS every where 90 degrees from the position of the oye. It derives its name from the fact that it results from the intersection of two solids, the cone and the sphere. Its principal properties are the following: i. The pro- jection of any circle on the sphere which does not pass through the eye is a circle; and circles whose planes pass through the eye are straight lines, j. The angle made on the surface of the sphere by two circles which cut each other and the angle made by their projections are equal. 3. If C is the pole of the point of sight and c Its projection; then any point A is projected into a point a such that c a is equal to arc CA rx t^y^ where r is the radius o^ ,;,. sphere. From the second property it follows thn.i :y very small portion of the spherical surface and Irs projection are similar figures • a property of great importance in the construction of maps and one which is also shared by Mercator's projection. The astronomical triangle PZS can evidently be easily drawn on the stereographic projection. Z will be the pole of the point of sight. The lengths of ZP and ZS are straight lines found by the rule given above, and the angle Z being known the points P and S are known. The angles P and S being also known we can-draw the circular curve PS by a simple construction. ■ The orthographic and stereographic projections were both employed by the ancient Greek astronomers for the purpose of representing the celestial sphere, with its circles, on a plane. Gnomonic or Central Projection. —In this case the eye is at the centre of the sphere, and the plane of projection is Projections. 169 oro.WH ^''"'/''"^ ''^' '"^^''^ "* ^"y ^^^""^-d point. The projection of any pomt is the extren.ity of tlie tangent of contact. As the tangent increases very rapidly when th^ arc ,s more than 45", and becomes infinL at 'o't s evi hetisX^'^ '''''''-'' ^^""" '' ^'^'''' '-' ' ^^^ Globular Projection.^This is a device to avoid the d.s tortion which occurs in the above projections as w.' approach the circumference of the hemi phere In tie accompanymg figure let A C B be the hemisphere to be repre- sented on the plane A B, E the position of the eye, o' the centre of the sphere, and EDOC perpendicular to the plane A B M and F are points on the sphere, and their projections are N and G. Now the representa- tion would br perfect if A N : N G : G O were as A M : M F : F C. This cannot be obtained Fig. 39. iTr^U-/,' ^^" ^' approximately so if the point E IS so ^fi^that G is the middle point of A O and F he drlwt^F L ^ ^;- ?" '''' ^^^^' ^^ i-"-^ F O and ^xo.7r nearly;^^^': h^ } I "" l^'""' '''' °^ hoif /i • j;^-^»«:a»e*^ Vj ( half the radius and F L half the inscribed sqwive^-ih^^efefe- FL:GO::OC:OL butFL: GO::LE : O E .•.LE:0E::0C:0L consequently, LO : O E :: C L ; O L, or O L^'-O F r r but O L2=F L3=n T ir ^' "^ '-' i- — ^ E.C L ^ L -U L. L C, .-. O E. C L^D L. C L or O E-=-D L that is, E D— O L II IMAGE EVALUATION TEST TARGET (MT-3) // 1.0 I.I ■^I2£ Hi W I4i 122 lU L& 12.0 u u IL25 i 1.4 1.6 Photographic Sdences Corporation 23 WEST MAIN STRUT WEBSTER, N.Y. 14580 (716) 872-4503 '41!^ '^' ■^ „* * K^ 170 Projections. The above projections are seldom used for delineating the features of a single country or a small portion of the earth's surface. For this purpose it is more convenient to employ one of the methods of development. Mercator's Projection is the method employed in the construction of nautical charts. The meridians are repre- sented by equi-distant parallel straight lines, and the parallels of latitude by straight lines perpendicular to the meridians. As we recede from the equator towards the poles the distances between the parallels of latitude on the map are made to increase at the same rate that the scale of the distance between points east and west of each other increases on the map, owing to the meridians being drawn parallel instead of converging. If we take / as the length of a degree of longitude at the equator (which would be the same as a degree of latitude supposing the earth a sphere), and /'that of a degree of longitude at latitude X, then /'-/cos X, or /'.•/:: i : sec L Now /.' : / IS the proportion in which the length of a given dis- tance in longitude has been increased on the map by making the meridians parallel, and is therefore the pro- portion in which the distance between the parallels of latitude must be increased. It is evident that the poles can never be shown on this projection, as they would be at an infinite distance from the equator. If a ship steers a fixed course by the compass this course is always a straight line on a Mercator's chart. Great circles on the globe are projected as curves, except in the case of meridians and the equator. In this projection, though the scaie increases as we approach the poles, the map of a limited tract of country gives places in their correct relative positions. The Ordinary Polyconic Projection.— In conical develop- ments of the sphere a polygon is supposed to be inscribed in a meridian. By revolution about the polar axis the Projections. j^j polygon will describe a series of fru^ums of cones. ~Tf the arc of the curve equals its chord the two surfaces will be equal. In this manner the spherical surface may be looked upon as formed by the interse'^tion of an infinite number of cones tangential to the surface along succes- sive parallels cf latitude. These conical surfaces may be developed on a plane, and the properties of the resulting chart wWl depend on the law of tUe development. ,J^^ ^'fT'^ Polyconic is a projection much used in the United States Coast Survey. It is peculiarly appli- cable to the case where the chart ambraces considerable difference m latitude with only a moderate amplitude of longitude, as it is independent of change of latitude. Before describing it it must be noted that whatever projection is used the spheroidal figure of the earth must be taken into account, its surface being that which would rnninT f ^^" revolution of a nearly circular ellipse round the polar axis as a minor axis. In the Ordinary Polyconic each parallel of latitude is represented on a plane by the development of a cone having the parallel for its base, and its vertex at the point where a tangent to a meridian at the parallel in tersects the earth's axis, the degrees on the parallel pre- serving their true length. A straight line running north and south represents the middle mt ridian on the chart and ,s made equal to its rectified .a . according to scale.' Ihe conical elements are developed equally on each side of this meridian, and are disposed in arcs of circles described (in the case of the sphere) with radii equal to the radius of the sphere multiplied by the cotangent of the latitude. The centres of these arcs lie in the middle Sude" P'"'^"'''^'^' ^^^^ ^'^ ^""'^'^ 't at its proper These elements evidently touch each other only at the middle meridian, diverging as they leave it. The cqrva- f I 172 Projections. m tureofthe parallels decreases a. the distli^^TT^^e stlTgitTnr'' ''" '' '^' '^"''°' ''^' P'""""^' ^^^^'"^^ ^ To trace the meridians we set off on the different parallels (according to the usnal law for the length of an arc of longuude) the true points where e.ch meridian cuts them, and draw curves connecting those points To allow for the ellipticity of the earth we must use for the radms of the developed parallel N cot /, vvhlre N— a nn!'!?. *^' '^""*°"^^ ^^d'"«. ^ the eccentricity, N the z:^:^Z2^ atr "- ^- -' ^ ^'^ '-^^ ^^ It is evident that the slant height of the cone— say r— is N cot /, and that the radius of the parallel on the spheroid is N cos /. The length of an arc of n" of a parallel will be «°-^-o N cos /. In practice, instead Fig. 40 of describing the arcs of the parallels with radii, it is more "Sfufjf TfT / ^^<^'^^^ f the" ^) that h r 'P'f P'"^"^' ^^ ^°* ') -"d the angle itf) that this radius makes with the middle meridian. 1 (I) (2) Projections Take the origin at L (Fig 41) the point of in- tersecfion of a parallel with the middle meridian- tlie middle meridian as the axis of >.; and the per- pendicular throngh L as the axis of a:. Then we shall have for any point P whose latitude is / and longitude from the meri- dian «' ^=-YPsin/?=»Ncot/sin""■- thepointtoLproieceS !;:"•;,'.! P"^"^' "■""sh 'ion (., wi„ becUr;fN^r ter ±T"'"' ^''"^- 174 Projections. m From the above equations tables may be formed for the construction of charts. Imj^'. 42 shows the geometri- cal relation between tlie angles ff and n. This projection, when the amplitude in longitude does not exceed three degrees from the middle meridian, has the fol- Fig. 4^ lowing properties. It distorts very little, and has great uniformity of scale. It is well adapted to all parts of the earth, but best to the polar regions. The meridians make practically the same angles with each other and with the parallels as on the sphere. Angles are projected with little change. The great circle or geodesic line is projected as a straight line practically equal to itself. '1 V f( CHAPTER VI. TRIGONOMETRICAL LEVELLINO. TO PIND THE HHICHT OP A POINT B ABOVE A STATION A. In the accompanying figure O is the centre of the earth, AC is tangential to the earth's surface at A, B' is the apparent position of B, owing to refrac tion. CC is the correction for curvature, or-^^^ , where K is the horizontal distance of B from A, and R is the radius of Fig 43 the earth ; both in feet. BB' is about 0.16 CC ' ACB n4y be taken as a right angE?/n^^ , the arc AC . and the straight line AC, ait/ft^. We shall BC=K tan B'AC + CC-BB' ==K tan B 'AC + 0.00000002 K^ ''^ where B'AC is the observed angle of elevation of B This ^ formula supposes that AC B is practically 90°. ff 'the dt 176 Trigonometrical Levelling. tance is so great that this is not the case we shall have in the triangle ACB ,sin BAC BC=K- sinB To find the angle B, we have in the triangle AOB, B=-i8o°-(0+BAO) =180— (0 + 90° + BAC) — go°— (O+BAC) Hence, sin B—cos (O-f BAC) BC=Kx ^^"^^^ cos (O + BAC) And BC'™BC + CC'— BC + K" 2K (whore R ia tlit mdiuj t>f Lthe eafth-in-feet.) The angle O, in minutes, is 0.0001646 K. and -p- IS 0.000000023936 K* REFRAC- RECIPROCAL OBSERVATIONS FOR CANCELLING TION. If we measure the reciprocal angles of elevation and de- pression of two stations— in other words, if at each we observe the zenith distance of the other— we shall get rid of the effects of refrac- tion. Let a be the angle of elevation of B at A and ^ the angle of depression of A at B. Then BC'=-K X -^^S'*±R^ cos I («+^-fO) •* If the zenitli distances areobserved call them rocal observations ought to be simultaneous in order that the effects of refraction may be as nearly as possible the same for both. In problems of this kind we ought, strictly speaking, mstead of using the mean radius of the earth, to take the normal for the mean latitude of the stations. The following geodetical formuloe are used for more exact determinations. In addition to the letters used in the foregoing problems we have a the known altitude of the lower station ; N the normal for the mean latitude ; M the modulus of common logarithms; and r the co- efficient of refraction. I. TO FIND THE DIFFERENCE OF LEVEL BY RECIPROCAL ZENITH DISTANCES. Log.diff. of level=log |k tan J {d'—d)\ M + 1j^« ± ~^f K tan J {d—8) -|- N 2 N ■* ^ 12 N' rKS 2. TO FIND THE DIFFERENCE OF LEVEL BY MEANS OF A SINGLE ZENITH DISTANCE. Log. difT. level = log. M K tan \r- J=1JL K I ( 2Nsini" J ] K M tan{^--|=?:r;K]"'"N» V 2 N sm I J The third term is positive if a is less than go*. K» Trigonometrical Levelling. 179 3. TO ASCERTAIN THE HEIGHT OF A STATION BY MEANS OF THE ZENITH DISTANCE OF THE SEA HORUoN In this case, when possible, different points of the hori- .on should be observed on different days and the meln of the whole taken, the state of the tide'being also nofed 1 he formula is Log. aI.i.ude=IogA(^) •+,„,. (,_,„.,, , M / sin I" ) « ^ The angle ^— go' is in seconds. The last term may generally be neglected. levll'^h J°"°'''';^ '' an example of finding the difference of level by a single zenith distance. an^A' 1.'"?-^ '^' lower station («) was 1000 yards, and h or the height of the instrument 5. ' . J.fi' '^^f °";^! <^istance between the stations (K) was r'UX',0^'' '-'''' '''^^- °^ ^'^ -P- -tio^n First, to find the value of the angle 5. Log /t=o.69897 Log sin A =9.99984 Co-log K^5.2378o Co-log s'n i"=5.3i443 Log h sin A K smT"~^-25i04=Iog ly.g Therefore ^=88° 24' ^y".8 I — 2 r '2N sinT' Next, to find the value of the angle K Log I — 2 r ^-jf-^lj^^ =8.13252 Log K =4.76220 "iV^" K=88" II' 53".i 2.89472=log 784".7=o^ r^' ^"^y I i8o Trigonometrical Levelling. Thirdly, value of the difference of level. Log K =4.7621004 Log tan 88" ii>53".i =1^022427 Log 1st term=3.2599557 2nd terni= 691 3rd term= -I-627 4th term= g Second Term Log M = 2.8353 Log a =3 Log 2nd term = 58393 Log. diff. level=3-26oo884=log 1820.07 yards. Third Term Lo8-'^ = 25383 2 N log 1st term =3' 2599 log 3rd term =.5 7982 and term =0.0000691 I 3rd term =00000627 Fourth Term. 2 N log KS =9'5244 "-"^-xTn-^^s? ZENITH log 4th term =2 963 1 ' 4'h 'erm= 00000009 REFRACTION, die. TO FIND THE CO-EFFICIENT OF TERRESTRIAL REFRAC- TION BY RECIPROCAL OBSERVATIONS OF DISTANCES. Let A and B be two sta- tions, and let their heights (ascertained by leveHing) be h and /»'. Consider the earth as a sphere, and take O' its centre. Call the radius r and the angle AOB v. Let Z be the true zenith distance of B at A, viz., ZAB, and Z' that of A at B or Z'BA. The dotted curve shows the path of the ray of light. A' and B' are the apparent positions of the sta- tions. The co.efflcient of refraction is the ratio 'of the differ ence between the observed and real zenith dSance at' I I Trigonometrical levelling. either and station observ m.,,M , theanplet;. Thus, if A is the co-effick... 1, J^';;«_oJ>^e'-ved .enith distances, wc have k equal to i8z lent -or -z V But these are not always the same. In the triangle AOB we have HZ' + Z)=9o-+ V V tan -tan \ C/J—'A)^ These equations give '£ and Z. V :' + 2 y + A If we substitute for tan ^- the first two terms of its ex- pansion m-«a«,. the second equation may be put in the A'— A=s tan ^ (Z'— Z) 1 1 + h±lL ^ _l!L I where . is the length of AB projected on'thV'sea level. the sirnnlT'^'''"* ""^ ''^'■^'*'°" "^^^ ^'^° ''^ ^^^^'^^^ ^O"^ o A and R'"Tr :^,^'P'-°^^"y-bserved zenith distances ot A and B without knowing their heights. Thus : Z=zz-^kv, and Z'=z'-\-kv •: z + z'+2 kv~i8o° + v .-2 + ^'— i8o" or 1—2 k=' V sea'ul'oSn '°"fr''"' '' '''^''- ^"^ '^y' ^^-'"^ the sea It ,. .o8og, and for rays not crossing it .0750 and not to be expressed by any single law. In flat ho countnes where the rays of light have to pass nea the .round and through masses of atmosphere of diffrni dens.t.es the irregularity of the refraction is very grelt so much so that the path of the rays is somedm s -nvex to the surface of the earth instead of concave In Great Bnta n the refraction is. as a rule, greate t Tn tl e early mornmgs ; towards the middle of Se day it d" creases and remains nearly constant for some hours, in- creasmg agam towards evening. CHAPTER VII. THE US. - ;™ sx^^./.^.-.is-- ^- The spheroidal form of the earth causes the force of gravity to increase from the equator towards the poles, and th,s force may be measured at any place by means of the oscillations of a pendulum. "'eansoi If we had a heavy particle suspended from a fixed poin by a fine ine:;tensible thread without weight we should have what is called a sunpie pendulum. \i\Z pendulum were allowed to make small oscillations (of not more than a degree in amplitude) in vacuo, and in a ver- fo'mSa"'' '' '""' °^ °^"'"^tion would be given by the Where t is the number of seconds, / the length of the pendulum m feet, and g the force of gravity. nJ^r^°'?' *^^'"^^ ^^ ^°"«t^"t. if there were another shtldZe '^" '°"^' ^"' ^^^^^^^"^ - ' — ^^' - t : t':: 4// : 4//' or, if the time were constant and g changed to g', Pendulum Observations. 1S3 HE e of Dies, IS of xed we this not 'er- the he ler ive (2) or if/ were constant and g variable ^ g ^ g d . / ^^^Ti'' '^'^, ^^^ ""'"^"^ °f oscilIationsan'4^rime) f o. t and ti, then, » ' :n:U"t' i -«« "mey :: f// : \/l' From (2) we have, 5-'=-^ g- -VTa^ (3) To find the value of ^/ we can either ascertain by measurement the length 'a pendulum that makes a cer- TZ:Z °/.°^^'"^^-- •" - ^-en time, or we can use a pendulum of'nvanable length and find .,' from equation Siesfrpr^tt'^ ''-' '^^" -''' '-' ''^'-'■- ^'^ _ A simple pendulum as described above is, of course an nnagmary quantity, and all pendulums actually u da" what are called "compound" pendulums. But it is pos- sible to calculate the length / of a simple pendulum that would osallate in the same time as th'e compo "" one that x" ofthe' '^fZ ;' ^'^ "^^"^- °^ oscillation"' that is of the pomt which moves in the same manner as th^: ol?:f"'"" '''^ ^^^°'^'"-^ -- -''-"da tha point, thus constituting a simple pendulum. The centres of osc.llat.on and suspension are interchangeable and If a pendulum is suspended from the former th^ latter becomes the new centre of oscillation. The compression of the earth is calculated thus ■ If , IS the compression, ^. the latitude of a station, gtl'e force the':::^' r.v'' ^^t^^' ^' *^^^ ^^ ^^^ station :„;: the ratio of the centrifugal force at the equator to ^ we have, by the formula known as Clairaut's Theorem S =S [i+(f m-c) sin2 ^J and. since ^'-«-i. ,, if „ ^3 ,^^^ ^^^^^^^^, ^^ oscillations in : I Pendulum Observations. fe""'"^^' '"« '^^^^^'^^^^^^^^n^^^n^. n I' />. (4) [being the length of the pendu um at th. . /^^ bemg known, and n«^ or //'beinTfounih "^ °'"- '" we at once get the value nf f ^^ experiment, get tfte value of c from equation (4) or (5). (that is. a ;::du, Itcitfn, i T'T''' P^"'"'""' ferent stations, consisted o "a' pherT'o; 7T'^ ""'''''■ pended by a fine w.V^ ... u^ ? P'^^'""'" sus- whichwasakn^fe .r 'f ^'^ '° ^'^^ "PP"^^ «nd of Plane Th^ u /l°^'^''' resting on a level a<^ate seconds pendX^nrlL "'"'"=" '."^ ''"S"" "^ '^e between thp fu7«l^ ='»u'ng weight. The distance dul„^ the two edges gave the length of the simple pen- two L'ot'^xf.r''^'^'"'"""" " ""' ■■' "--- Pendulum Observations. 185 using a sliding weight, one end of the bar of which it consisted was filed away until the vibrations in the two positions were synchronous. In using the pendulum it is swung in front of the pendulum of an astronomical clock, the exact rate of which is known. By means of certain contrivances the number of vibrations made by the two pendulums in a given time can be compared exactly, and the number made by the clock being known that of the experimental pendulum is obtained. Certain corrections have to be applied. One for changes in the thermometer, which lengthen or shorten the pendulum : a second for changes iu barometric pressure, which by altering the floatation effect of the atmosphere on the instrument, affect the action of gravity on it ; a third for height of station above the sea level, which also affects the force of gravity, the latter diminishing with the square of the distance from the centre of the earth ; and a fourth for the amplitude of the arc through which the pendulum swings, which, in theory, should be indefinitely small. The number of pendulum oscillations in a given time has been observed at a vast number of stations in various parts of the world, and in latitudes from the equator to nearly 80°. The most extensive series of observations was one lately brought to a close in India, the pendulums used in which had been previously tested at Kew. The general results of all the pendulum experiments gives about 292 : 293 as the ratio of the earth's axes, which is the same as that deduced from measurements of meri- dianal arcs. f1 / • i n r . '7'