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N.B.— This Society, as a body, does not hold ItHelf responsible for the toots aud ophiiona stated In any of its publications. THE ERRORS OP LEVELS AND LEVELLING. PAUT II. -LEVELLING. By Prop. C. H. McLeod, JIa. E., M. Can. Soo. C.E. To be read on Thursday, 18th December. In considering the errors of levelling, reference will only be made to such operations as are ordinarily included under the name of spirit levelling, omitting any reference to trigonometrical levelling, in which the spirit level play.s an equally important part. It will first be proper to remark that spirit levelling may be broadly divided into two classes : The ordinary levelling undertaken in connection with railways, canals, drainage, etc.; and "geodesic" or "precise" levelling, where the highest attainable accuracy is sought in fixing the elevations of inland points with reference to a common mean sea-level datum. Nearly all European countries are carrying out such systems of levelling in con- nection with trigonometrical surveys ; and in the United States many thousands of miles have been run under the direction of the Coast and Geodetic Survey, the Lake Survey aud the Mississippi Biver Commis- sion. Some few hundreds of miles have, T believe, been levelled in Canada by precision methods, but I have not the details before me. The instruments employed iu spirit levelling may for convenience be divided into three classes, — the dumpy, the wye, and the precision level. The form of the dumpy and wye as employed for ordinary levelling need not be described here. The precision level appears under a variety of forms. Those which have been used in the best European work and in America are of the wye type. The level tube is detached and used as a striding level on the collars of the telescope. One wye is move- able in a vertical direction under the control of a micromet<.'r screw, in order to obtain the small liual adjustment necessary to bring the deli- cate level employed to a central position on the scale. There is a wide range of scale value in the levels employed in the two classes of work, and also amongst the individual instruments used in each class. This divergence is much more marked in the ordinary levelling instruments than in the geodesic instruments. In illu.stration of this point, the scale values of the levels (as they came from the hands of the maker) belonging to MeGill College are given in the table below. The scale values employed for geodesic instruments are usually within the limits of 3" and 6" per division of 2"'™ ; equal respectively' to 3.75" and 7.5" per div. of 0.1 in., the length of the division on the levels refierred to below. Scale Vai.uks at 65° Pali. Kind of Instrument. Maker's Name. Scale value per (liv'n of 0.1 in. Optical powers. 12 in. Wye level. HainnuT.'^lev. 75" 18 und 26 14 in. Dumpy level "A." 1),. 21" IH " 26 14 in. (io " "B." Do 26" 20 " 25 14 ill. do " Stanley. 27" 20 " 25 14 ill. do Trou)rlit()ii & Sill, 111-:. 14" 18 " :!0 IHin. Wve. Hiitr & Berber. rr :t7 The optical powers in this li.-^t will be referred to hereafter. Before proceeding to a consideration of the errors of levelling, it will be necessary to review briefly the methods of adjustment in the several forms of instruments, and in this connection certain defects of construc- tion will most conveniently be referred to. The object of all methods of udjustmeiit is to bring the line of sight purullul ti^^) the lino tangent to the inner Durface of the level tube at its aero point, — this line will in this paper be referred to aa the " bubble- axit*. " In the dumpy iIiIh condition eun only be reached by the direct luuthotl of reading the rod on two " pegH, " the diilereuouin the level of which is known, one ])oint being near the instruuient while the other is several hundred teet distant, and shiftin)^ either the position of the cross hairs or ot the level tube, until the readings give the true difference in level. The only error in this method is the slight one due to the curvature of the earth, amounting to about J_th of a foot in 100 600 feet, and to one-quarter that amount in 300 feet. 'I'iie method usually employed to obtain the difference in level of the pegs is by setting the instrument at a point midway between them. The errors of the readings then being the same, the difference is the true one. A second method of determining this difforenoe of level, which is not, I believe, so generally practised and perhaps not known to all Engineers, is as follows; — First set the instrument over one of the pegs, so that the eye end will just swing clear of the rod, held vertically on the peg. When the instrument is level, view the rod through tiic telescope, with the eye at the object end ; the centre of the small cir- cular portion of the rod thus seen can be accurately estimated and its position read by the aid of a pencil or knife edge held against the rod. Then read the rod as usual on the distant peg. Now move the instru- ment over the second peg, and obtain a reading of the rod as in the case of tlie first peg. Then read the rod on the first peg which is now the one distant from the instrument. If we call the difference of the rod readings in the first position of the instrument m, and the differ- ence of those in the second position n, then the true difference in the elevation of the pi^gs is Results thus obtained, being from four separate readings instead of two, and being independent of the accu- racy of any horizontal mtMisurement, are suf-ceptible of greater precision tlian the usu;il nictlmd. The instrument and rod are now in position for the necessary adjustment, and the distant rod reading will be corrected by tlie above amount. It is customary to adjust the wye level by the indirect method. The line of .sight lieiiig examined for coincidence with the axis of the collars by revolving the telescope in its wyes, the bubble-axis is then tested for parallelisin with the bearings of tlie eolhirs by turning the telescope end fur end in its wyes, and the necessary corrections ap(ilied. This lueihod is based on the assumption that the collars oi the telescope are true cylinders, au is not an uncommon cir- cumstance, in my experience, amongst users of wye levels, to find the fulfilment of the fir.«t portion of this test — the revolution of the telescope in its wyes — accepted as :i guarantee of the perfection of the whole. Such an error of judgmi iit is of eour.se impossible with persons ac- quainted with the theory of the instrnmeut ; but alas, the holder of the position of ''Engineer" in this free country of ours may readily be a touch above theoretical considerations. For convenience only is it desirable that the bubble should remain central while the instrument rotates about its vertical axis. In the dumpy this adjustment must be made before, and in the wye after, the essential Jidjustmenls have been completed. It is commonly stated in defence of the indirect methird, that there is a saving of time as compared with the direct or "peg" method. Such is however not my experience. The time-consuming portion of the work is the mak- ing of the necessary corrections to the level tube and reticule. In the essential adjustments by the indirect method both moy require correction, whereas in the direct method only one of them — preferably the reticule— should be moved. We have then in comparing the rela- tive convenience of the two, on the one hand the matter of setting up the instrument once and making two or three rod readings, and on the other the complete adjustment of the level tube. I had rather set up ten instruments and make as many rod readings, and could do it 2 in iess tiniu, tlmn niljuHt oin' bulililc tube. This indirect lurthod wiems to mu to liavc, tor tlii' ]>ur|i(isrs of ordiniir}' levelling, no cnc point to coiumcnd it in iirdirciifc t'> tiic dinvt mt'tlioil, itnd ImHtlic ditiadvau- tttsro that it does nut ^xniiiiintci' unrrcct iiiiju>tuu'nt, unk'HH thu pivotx liuvi!, afttT oar'Tni i.'xanilnat ut in the hand of man Ms an old and shaky wye level. I submit that in the interest of good vork, if not of economy, the construet-on of the wye form for ordinary levelling should be abandoned in favor of the more compact and rigid dumpy. The modification of the dumpy, such us Cushing's reversible level and other deviations from the type i'orni, do not, for similar reasons, com- mend themselves to me. Tlie adjustments for coincidence of the optical axes of the objective leiiH and the eye-piece with one another and with the axis of the tube, provided for in the wye but not in the dumpy, are really makers' adjustments, and except in the case in which the* axis of the objective slide makes nn angle with that of the tube arc not essential to correct work. A very important point to be at- tended to in the construction of all levels, and in the examination of them from time to time by the Engineer, is that the object lens is not loose in its cell, and that the cell i> not loose in the telescope tube. Both these defects are apt to occur through time. The tightening band at the back of the cell should screw into place in order to per- mit such a defect being rectified . An instrument having a loose ob- jective is impossible of adjustment and fatal to good work. The accompanying cut taken from Fauth's catalogue, represents an instrument of the form used in geodesic work. The pattern employed by the United States coast and Geodetic Survey difiers slightly from this instrument, which is the one presci il^ed by the International Geode- tic Commission, held in Berlin in 18G4, and used in this country on the Lake and Mississippi River^surveys. The followinif description o ' tii • iii-ifniiii 'iit is taken from Johnson's theory and practice of survey iuLi; — " The bubble is enclosed in a wooden ease (metal case in the cut), " and rests on top of the pivots or rinus : it is carried in the hand " when the instrument is transported. A mirror is provided which " enables the observer to read the bubble without moving his eye from " the eye-piece. There is a thumb-»crew with a very fine thread " under one wye, which is used for the (inal levelling of the telescope " when pointed on the rod. There are three levellinsi-sercws and a " circular or box level for convenience in setting. The telescope 3 " bubble is vcr/ dolical 3, one division on the sciilo corresponding to " about three seconds of are. The bubblotube is chambered also " thus allowing the litngth of the bubble to bo adjusted to different " temperatures. The niii^'nityin^; power is about 45 diameters, " Tliere are three horizontal wir. « providid, set at such a distance " apart that tiie wire interval is about one-iiundredth ft" the distance " to the rod. The tripod lif;s are covered with wliitf cloth to dim- " inish the disturbing effects of the sun upon tliem. Tiie level itself is " always kept in the shade while at work. " The levelling-rod is made in one piece, three metres long, of dry " pine, about four inches wide on the tiioe, and strengthened by a •' piece at the back, muking a Ti*l> 'P-J cross-section. The rods are " self-reading, that is, tiiey are witliout tir.;ots, and are graduated to *' centimetres. An iron spur is provided at tlie bottom, which fits " into a socket in an iron footplate. The end of the spur should bo " flat and the bottom of the socket turm:d out to a spherical form " convex upwards. A box-level is att.iehed to the rod to enable the " rodman to hold it vertically, and tliis in turn is adjusted by '' means of a plumb-line. Two handK's are provided for holding the " rod, and a wooden tripod to be used in adjusting tiie rod-bubble. ■' The decimetres are marked on one side of the graduations and the " centimetres on the other; all figures are inverted since the telescope " is inverting." In the use of such a level according to the methods of precise ovelling, it is necessary to determine the instrumental constants : — 1. The angular value of one division of the level tube. This may be found in the field by sighting on a rod at a known distance, taking readings of the bubble along the whole length of the graduated part of the tube, but it is more satisfactorily diterniiued on a solidly mounted level trier. 2. The inequality in the diameter of tlie telescope collars. This is found by reading the striding level on the collars in the two reversed positions, thus eliminating the error due to the unequal length of the striding level legs and obtaining the true inclination of the surface of the rings. The telescope is then reversed in its wyes and tlie levelling repeated. The difference in the two inclinations, divided by four, gives the angular value of the error or correction sought.* 'f he fol- lowing observ:itions are given as an example : — Level roadiii):^. il Objoet end. Striding Level direct Eve end 7.4 rever^jed 8.0 !).0 8.4 Muiiu8=7.7 and 8.7 7.7 2)1.0 Inclination of upper surface of eollors.=0.60 Telescope reversed in Wyes. Level reailings. Object end. Eye end. Striding level reversed 5.6 11.0 " " direct 5.0 U.G Meang=5.H and 11.3 5.3 2)6.0 Inclination of upper surface of collars ==3.00 Difterenee of readin j.'s =3.00— 0.50 =.0.625 4 " '4 The value of one division of the level \vas=o". Ilem-e, correction in •secondsof arc = 5 x 0.625 = 3. 12, and the eye end is tlio largest, requiring a negative correction to all rod readings. 3. The ratios of the portions of the rod intercepted between the three wires and the distances of the rod. This for the double purpose of obtaining a measure of the length of -iglits and furnishing a check on the readings. The adju.stments — making due allowance for the fact that the level tube is moveable — arc the same as those for the ordinary wye. Since, however, it is impossible to do anything exactly, and in the best work the smallest errors should be provided for, the after-treatment of the adjustments is essentially different. They are reduced to as near zero • See Chauventel's Astronomy, Vol. II, p. 153. as posBtblu, and thi> ouUtundin^ urrom detcrminod ai foUoWR : — 1. The diifurenoe io the iivernKU oi' thi; thruc- rod rLadin^a in the two reversed positions of the tt'lcscupo tube ^ivL'H twio<' the ooUiniation error of the mean lino of sight, at the known distance of the rod. 2. The inclina- tion of the bubblt-axlH to tiic top surface nf the rings is found by taking a scries of readings of thi* bubble in the reversed positions, the average of the mean differences at the eye end and at the object end, for level direct and level reversed, gives twice the correction required for the inclination of tlut level tube. Thus, in the example already given, the ditferenoe is O.ti div. and the correotion 0.3 x 5.0= l.'^SO. These determinations ure made at the beginning and the end of each day's work, and their resulting values combined with the correction for inequality of collars, applied, in the reduction of the notes, to all rod readings. Methods of work. The method of using these instruments difiers from ordinary work only in that there are three rod readings for each setting of the rod, and the reading employed is the mean of these. In some oases the bubble is kept central by means of the micrometer screw and the reflecting mirror ; and in other cases only approximately so, tlie ends being read and recorded by uu assistant. A correction is, in the latter case, afterwards applied to each rod reading for the inclination of the line of sight. The three readings of the rod in each position should always be compared before the level is moved, in order, if neces- sary, that any doubtful reading may be cheeked. Self-reading rods usually graduated to centimeters and estimated to millimeters are employed on the Lakes and Mississippi River surveys, and in most of the European work. On the Coast and Geodetic Survey target rods are used, and the method of work there pursued is much more elaborate than the foregoing. The methods of levelling are sometimes described as nngle or double levelling, according as single or double back and fore sight are taken. The latter has been the practice of the Coast Survey, and is a self- checking system — in so far at least as the readings, are concerned — really amounting to two lincs_,of levels in the same direction. The single system is however the more generally adopted. There is a decided economy in time and in accuracy of result in the use of two rods alternating with one another on turning points. In reviewing the construction of levelling instruments, the principal points to be held in mind as conducive to the best results in the various departments of work would seem to be. First, — stability of construc- tion ; second, properly constructed levels of sufficient delicacy ; third, adequate optical power for the purpose in view. The first condi- tion should be secured by a proper distribution of the metal, for the attainment of a maximum of strength from a minimum of material. The second and tliird are to a certain extent interdependent. For railroad and canal work from 10 "to 20" per division of 0.1 " would seem to be limiting values for the scale values nf levels. Levels having a scale value of 12 " are entirely satisfactory on ordinary work, while those beyond 20 " I have considered us lacking in sensitiveness. It has always seemed to me better to have an over-sensitive than an under- sensitive level, admitting the difficulty of keeping the former central. Increased optical power is of course secured at the sacrifice of light and definition. The powers demanded foe our instrument are however much within the limit of uood lenses, under the ordinary conditions of illumination. Referring to the list of instruments already given, the practice of the makers of dumpy levels seems to be keep in the neigh- borhood of 25 for the higher powers for 14 in. instruments. The object lenses of all these instruments would I think stand higher powers. With a good object glass of 1 .5 in. aperture there should be no difficulty is using a power of 40 under the ordinary conditions of seeing. In the smaller apertures of the wye levels the matter of loss of light be- comes serious. But the makers of wye levels seem altogether born to perversity, — having decided to use an object glass of small aperture, they must of necessity add to this a four lens eye-pitce, as if it were any advantage to get an erect image. The four lens, inverting eye- piece has no advantage over the ordinary erecting eyepiece, and its use results in serious loss of light, giving a comparatively indistinct image ^-one of the most fruitful sources of error in levelling. 6 TliuiiM) oi tilt' mirror mi luvi^l lulx'^ to viuw tlic bubbli* \n not in Hurh favor witli I'li^iii.'orM im it hIiuuM bf. The uhuuI |ii'(ic'chm of ri-Htling the rod wbilc xtiiniiin^ in ii diO'iTcnt |Hiii,tiiiii (roiii ttiiit in wliich tbv intttruiucut wuH iovrll(>il luiiHt ii rcHsurily introduui' rrmn* iit'a Horiouit naturv. When tlin mirror iH umil it Hhoiiid uiwityM bo |iluc('d ut thu ■auiu inulinution, iind tlitt obNiTvor Hhniil i by trial, liMxint'd by anothi'r pvrHon, obtain that poHilion of tin' eye in wliicii tlic bubbji- may bu vicwtjd without parallax, and tVoni whiuli he will alway!< at'tt^rwardw obsurvo it. It i8 UHual to uhiMit'y souruoi* of orror somewhat ad follows t—l, instrumuutal rrrorit ; 2, urrorit from uiiNtabhi Hupportx ; 'A, urrorH of ubourvatiuD ; 4, pt'inoual crrorH ; 6, atmu.iplii'rio rrrorn. Tho chief iiourcu u{ v.rror from Inufnimnitnl caus/s i^^ no dtmbt due to oithcr apurmanentor temporary laulc of paralliliMii bttwucn tho lino of aight aud thu bubblu-axiH. No adjuNtmeiit can be ptrfoct, and I'Vt'n if perfect under atatt'd conditions is liable to chanu*' under other oondi- tiou)), Tiiu uauHuH which mi){ht produce a ehaiixe in adjutttmeiit due to thu influeneu of varying; temperature un the ulaoM of tho level haH been diHCUssed iu the tirHt part of thin paper. A ti'iuporary cliange may result from the unequal heating of the mi tul of the iniitrunient, which would probably take place in bright weather when levelling, in a direction towards or away from thu sun, The direct action of tlu; sun should be avoided by the use of a shade — a heavy canvas umbrella is used on geodetic work. In order to obviate the effect of any possible change in the scale value of the level, rcadinxs should never bo tiiken when the bubble in at any considerable di.^tance from central. If the bubble is sluggish there is a possiblo source of error in its bein^; read before final settlement. This latter difficulty can be overcome, with a cham- bered level, by avoiding; the use of short bubbles. Errors arising from defects of adjustment are of course completely eliminaUfd by equality of sights, except in such a case as that m 'nlioiied above, where, on account of the direct heating action of the sun, a change may take place between sights. Such changes are much more likely to affect the results of ordinary lev lling where there is an interval of time between centering the bubble and reading the rod. Errors of this nature are, though small for any given sight, of scious consequence through always being in the same dircutioii, or cumulative iu character. If there be, of necessity, a ditt'erence in the lengths of sights, he po.ssiblo error so introduojJ slioul I b; nrilr.ilizul by making, on the tirst opportunity, a similar ditference in the opposite direction. In the Indian Survey sights are made equal by chaining. Where, as for example iu crossing a wide river, a long foresight is unavoidable, the mitliod of rjeiproo:il levelling, explainjj in conn ^etioii with the peg adju'jtmont, should be i.'mploytsd. This also eliminates the error due to curvature of the earth. Where thero are two vertical wire- and the rod is road anywhere butween tliem. there may bu a sli.;ht error introduced tin ougli the lack of horizontality of the wire. This should be provided for in making the adjustment by swinging the telescope, when correctly levelled, around its vertical axis, and rocking the reticule ring until the wire is ob.served to continuously bisect a fixed point. Where no rod level is used, as is usually the case in ordinary levelling, it is however more important that tho vertical wiie should be truly vertical ; and where both cannot, in this case, be secured, the rod should always be read in the same position with horizontal wire. Wye levels should always have a means of preventing the t.'les(;ope from rotating in the wyes. Errors due to the rod will also fall under this class, and we should first look to the accuracy of its length and uniformity of its gradua- tion. Mr. I. W. Wright in his work on tin; adjustment of observations remarks that : — " An important source of error in spirit levelling, and one very "commonly overlooked, is the chan'j;e in length of the levelling road •'from variations of temperature. Fr^m experiments made by the " Prussian Land Sui vey, in which the rods were compared daily with " a steel standard, the following fluctuations in length were found for " four rods made of seasoned fir : — Rod 13, from Miiy 19 to Aug. 18 0.51 mm. per nietrt. 14 " " 20 " IB 0.40 " " " 9 " " 24 Supt. « o;n " " " 10 " " 24 •• « 0.43 " It in quite |)OHHilil(' that iriorN t'roui thin Miurocuiay lar^fly t^xooed " tho t'rrorn uriKiiij; from tin- Icvt'llinjt itwlf. I'lucli tii M purty iihould "theruforc he proviilml willi tho iiifiiiiH of mttkiii>j; u dmly .ompariMon " of thu rodrt UMud, with u ^tullUurd oi li'iigth. A nUv\ mutru nod a " mioromutur microHuope mouiiti'd on u htatid would bu all that would '• bo ueco8Hary." Thu maximum oxpaiiMion iibovi- <|UotiMl would amount to ,007 of a foot in thu it'n^:th nf n 14 foot rod. It will bu noted that the variation of temperature in not given. Kxperimenttt conducted by Prof. Van du Httude-liakhity/.eii on tiu^ ntiivex utud in thu Nuthurlands, fjive rc8ultH of much luHH magnitude then thu above. Hu found thu rate of ozpanHiun for MeuMoued Hr rodH to bi: 4.4 miora pur mutru, pur degree ccntigradu, and that other changes amounted iu uU to not more then .05 mm. pur mutru. For ordinary work the su-lf leading tjluscopo rod is almost univer sully adopted, on account of its poitability and cunvuniuncu in use- Tho rods of this form, though sold ax "Htundard " by thu makers, are, liable to bo moHtiiiuccurnte. Of tiie seven 14 ft. sulf-ruadiuK rods in use at McOill College, four are within .002 of a foot of buing correct one is .00(1 too Hhort, and two aie .014 too long, at C2° Foli. There are then two of thes«! rods, and surprising us it may suem, both from thu samu makers — a Loudon tirm of high ruputii — which ditt'ur between themselves to thu extent of .020 of a foot. Two 12 fit. target rods by different American makers are within .002 of thu correct length. Amongst a lot of five tclcscopo rods, recently measured in the ware- house of a dealer, one was found to be .020 too long at 14 feet and ,015 too long at 17 feet. In most of llie above eases the total error was roughly distributed throughout the length of the rod. In that last mentioned, and iu one of the College rods, the graduation was Romev.hat irregular. Where rods are properly constructo'd they should not bo influenced to any great extent by moisture. The utmost precaution should how- ever be exorcised to keep them dry. In thu Swiss levelling the errors due to temperature and moisture are stated as being "small, slow in action, and somewhat irregular. " Thu reports published from time to time of extreme changes from moisture are no doubt due to improper protection. Where rods are used under proper direction they are of coiirsc subject to careful comparison with the national standard, and any errors in their lengths allowed for. Errors arising from unxtahlv mpporH may occur through the instru- ment or the rod, and areusually of i eumuliitive character. The instru- ment may settle slightly between the reading of the back and fore sights, or the conver.-ic may occur, depending upon the character of the ground. Similarly, the turning points may settle or spring up between sights. In soft ground, settlement is likely tti oucur; while in stilf clay both instrument and rod — if tlie latter is supported on a peg driven in the ground — may spring back slightly. Pegs are preferred as turn- ing points for soft ground and for hard snil or roads, plates having a projecting knob or spherical hole for the rod, as already explained. The rod should never be niuoved from thu point until all thu read- ings have been made, and the weight of the rodmau should never be allowed to rest upon it. When botii instrument und rod move in the same direction the effects are additive, and the charactiT of the soil continuing the same over a considerable stretch of country, u large error may be accumulated. To obviate such a result, Colonel Walker adopted, in India, the plan of alternating the order of observations at successive stations of the itistrumeut, by reading the back staff first on one station and the forward >taff first on the next. The error may in part at least be eliminated by levelling between bench-marks in oppo- site directions. Duplie.ite levels in opposite directions between benches or along the whole length of line should completely eliminate it. This is indeed the sovereign cure for all errors of a cumulative character, Mr. Hirsch, one of the Directors of the Swiss precision levelling. 7 hui« Hhnwti tlint ihi- cnor liui! to xtttliuifiit, ntlur thiiiKN IxiiiK i<(|ual, In priiiiKi-tiiHiiil til till' li'ii'^tli iil'liiii' run. In iinlinitr)' IfvcllinK iipurn- lii>ll, till' cllltl'lli til' lit' lllr i:l'liUllil lllV''' I' till' Wiil'k ill II llifl'tTt'llt Way, the t'lvt of thi' iibmrvt-r ciiiii|irrHn (lie Lrriiiiml ncur tlu> tripod Iii^m, uml liJHpluct' th" liiic iil'siulil in tlir iimrvul nriiiin' ItrtwiMii lovt'lliu^ mid ri'ttiiiip till' roil, 'rooliviiiti' thi^. Oiilmi J I miili. r Iiuh v, i oiuiiiriid' iI that two of tlic li'^'rt III' iilwiiyM pliici'd piirail'l to tlut liiiiiof Ml^lit, 'I'ho Umi of tlui luirror.'ulri'iiily iioticril, wuuM iiUn r<'iiioV(> iIiIh Noun-i' of error. Till' lur;;rht Houri'i' ol iilnuttdliiiiinl iimrH in li liivil to bu duo to till' wiiiit of t'lirrful I'l'iiti'riii^ or rciidiii'^ the liiilihii', Kvtry levt'llur should know wliiit rod ri'iiditi); it* covt'iiil hy a raiiKr of oiuMJivl^ioii of hilt bulibli ikt a };ivi'ii diAtanuu of, >Ay lltl) fci't, iii ordor that hii may fully apprt'ciuti' till' t'lVi'cl of crrorHof tliiH kind. Whi'ti tlu' illiiiiiiMationof the two I'luN of the iuuiblu in ditfiTcnt, un error in ui'iiti'riiiu is ainiohl Hurr lo follow, ilirr.' brin^ u tciidi'iiuy to brin^ tliu bubblf too luiicli towards tln' li^lil. Krror ix uIho iiitruduucd throii;;h parallax, the bubble beiii;: viewed obiii|ue|)' to IIh length. It liu« recently been iiijieil by a German observer— Dr. Ueiiihertz— that the bubblett Nluiuld be viewed ill protlli'. L'learneHM of the glttHti and diiitincinusH of the ;:raduatioiii4 hav^' miiuli t'l do with the accuracy of bubble reudinp;!*, KrrorH of rod reudiiiK are more unmiuoii with a tar- get than with u Mjieaking rod. The bent elieek on the former Ih for both rodiiian and leveller to make iiide|eiideiit readin^H. Where three wireM are used, errors with Helf-readinu; lodn are of very rare occur- rence. The mean of the three reading.-* Ih aUo without duubt more acourutc than u Mingle reading on a turL^et rod. It dneit not by any me^nH follow that beeaUNe a target rod eads to .OUl of a foot that the reading is accurate within that limit. A ditt'ereiice in the illuiii- inatiou of the rod will also affect the relative accuracy of the readingH, and a line running east and west will probably show diflerent results, according as it is levelled in the forenoon or al'tirnoon. Under personal erroi-K we have merely to note that each obticrvcr ha8 his own peculiarities, which will largely atl'ect the resulting differ- ence of level over a great leir^ih of line. This is in fact the personal equation of thi' observer. To quote from the report of the Chief Engineer, U. S. A., for 1884 ; " These discrepancies vary with diffcr- '■ ent observers, and are not even constant for the same observer, :ire " nearly proportional tu the distance, and seem tu be independent of "the nature of the ground, the direction in which the work is done, •'the season or the manner of supporting the rod. ' The results of some recent levels on the Mississippi survey go to show that this personal equation may be somewhat evanescent, particularly with young obser- vers, and that e\ery line of levels .should Ik; duplicated in opposite directions by the same observer within tlii! shortest possibh' limits of time, in order to reduce the probability of change in the p.'r.sonal equa- tion. With skilled observers of long practice, this liubit is probably a constant from year to year. For t\ui i'Sxta of iitmosphurr in-nr.i I eaniint do lielti'r than (|Uoto from Profes.sor J. B. Joliiison, who was fir .some time engaged on work in connection with the Lake survey. (Van Nostraiids Mag. for Oct. 1883.) "Errors from this source may bt' classified as coming from : 1, "Wind; 2, Tremulousness ; 3, Variable Refraction. " 1. Wind generally .shakes the instrument, and make-> the holding " of the rod difficult or impossiWe. For two seasons 1 have used a " teuton windy days to protect the in.strument, aud withgri'at success. " Good woik can be done in this way so long as the rod can be held. " We also have large .square canvas umbrellas that can be set on the " ground to the windward of the instrument, and these effectually '' shield them in ordinary windy weather. " The tents used were wall-tents, 5 x li feet, and one 8-foot centre " pole. A square iron frame, H x 3i K-et, sewed into the canvas near " the top, formel the lateral .support ihere. It was held down by six " or eight steel pins, 18 inches long and ^ inch diameUu-, with flat " heads. These passed through iron rings sewed into the bottom. " There were openings for the line of sight and a flap for the observer " to enter and pass out with the instrument. The.se tents were made '' to be used on Gulf coast at a very windy 8ea.son, when one half tho 8 ' tioM would hftTA b«ti>n lout fVoin W\^^\\ wind* without th«iu. Thir rr)di liv Htlcki« hold in th<- Inind nmi braced men ' •i(Kin»( till' rod at tin iuikI)', rivitinu not thcntby litlml t'roiii their M)fki>l« in ' the I'uot pluti'H. '• m. Truiuulounni'*« ii cuninid by » iliffiTcnueof li>tn|H'ritturi< botwut'tt 'air und Kround, und itlwityn ouunrN in clrnr wnithi-r iilti-r thf itun iii ' u Cvw liourH liiKb. Tliii* iMkUMc^i tin- turi;i'l. nr tiKUrim en a i«puakint( ' rud, to Hppi'ur to uiovu up ami down, Kivin^ rim! to wimt ix known «« Jaiiuiiig or boiling. Thi-< s Hiinply riiu»'« of ' priH-'inioii. Till.' only ri'ni''ily iw to Nhorti'ii tliu li iigtb ol'Ni^ht ; but um ' thvruaruiMiuiuorrori tlmt luultiply diructly with tho iiuuibur ot'HiKhtK ' tukeu in a ^ivcii dintauvi', thoro ii iilixi u limit to wliirli thin rouiudy iiiay bf profitably uurrii'd. I do not think K HdviMibli' (o u^u Ni^btt loM than 100 tuut it' thu hij^huHt uufiiruuy U wiuj^hl, und purhapit neviT uiort than 40(t feet, ovon whm thi' utunwplii'r.! in pirfcotly • c'lt'ur und stt'udy. In cUrAV weutlur not nioitt limn ',i or 4 huun* a ' day uiui bo utilizod for tbc Im-'hI work. " 3. Variublo ruCrautiun ocourK whon tho Hun<>hinu Nuddcniy ooinim ' upur U'avi'h thu liae; tliiit buppfUM along thutdgL' of timber or uuditr " thu brow of a hill, uh when thu li lidly ilr uuiorgus ; " into the shado from tho huii'm movumi'iit, or on piirtiully cloudy dayx, " whuii thu Hun ii« iilturiiutuly coveiod und vicar. Whuii fiuiu thu lirHt *' Bourcu, it oucurt) about H a.m. und 4 p.m. It \h u peouliar phuuomu- '' uou, and in more common in winter tliiin in Humniur. Thu utmusphure " in uppuruntly 8teady und the Hight well taken ; but upon checking it, " thu nadiiig liaHulmnged, and may be observed to ehangu gradually or " 8udd nly, aud HomutinicM to i'i;cover a part or all of it)i original movu- I'luuut, whuu thu iuHlrumuiitM weru known to bu Htuble. I havu imud " thuMu ohauj^'H of ruadin^ amount to & iiidlimeturit, or 1-5 of an iuuh " in a diHtauce of 100 muters, or 328 feel. If thu utmoHphere in found " to bu in this condition, thu work should bu ! point, triumphantly brings it to you in bis huud ; or yut for thu leveller, who fails jHrsistently to distin- guish botwuen a IJ aud a ^. Luoking at the unavoidublu errors of luvellin^; in u more compru- ht'nsivu uiannur, wu may regard them as composed of thruu classes, — compensating errors, uujiulativu errors, and accidental errors. The first classes should he so manipulated us to ulimiiiute themselves dur- ing the progress of the, work. The second should be removed by the sume observer repeutlng thu work under us nearly us possible thu samu conditiouj, and iu un opposite direction. Levels checked only iu the sumo direction give fallacious results. Thu third aru the legitimate errors inseparable from uU observations, and are proportional to the square root of the distance. Thu errors of u properly conducted sys- tem of levels are usually oonsidered to be of this character, und their prcoision tested accordingly. The limit of error allowed n the Ft. U. 8. Coast and Geodetic Survey is 0.029 ./jjist. in miles. Lake Survey 041 y^Dist. in mi\^. Mississippi River Commission 021 J/Dist. in miles. between duplicate lines. The following interesting table of the results of levelling in Great Britain. India and Switzerland has been compiled by Mr. Wilfrid Airy, M. Inst. C.E. Average differences in a single mile of the results obtained by tw» obiervers, on ground of different degrees of inclination. 9 ClIAIlACTKK UK QltOrNI). UllK.VT Britain. Nearly level, very favorable oirciiM- Htoiiue-f ut' wealiier. Slightly uiiilulotiri)^,groiliefitsiiot exiiceil- ingriii lUO. OrndieiitH between 1 in 100 und 1 in 20. Gradients between I in 20 und 1 in 10. i Orndionts steeper limn 1 in 20. | l*'u(Pt. .(i2;i.s .o:n'.t .OAUO India. Foot. .0142 .OltJK .0208 .03S0 Switzer- land. Foot. .0125 .0148 .0183 .0308 .0410 NoTK. — The quontitieH in bold fiiee type are estimated from analogy iitlbnled by Swiss levelling, as no direct data could be furniphed. lu illu.stiatiou of tho hif^h degrx- ol' aeeur;iey iitliiined over long Jincs, tlio following is taken from the report of the levelling operations in India for 18GG, by Colonel Walker ; — jMaximi'.m Section. Length Miles. IN DiVERdENCE OF TWO UllSEllVEIlS. Foot. Calcutta to Tilliagarlii. 242 0.2(1 Tilliagarhi to Patka Oerouli. 34G o.to Agra to Putka Gerouli. .342 (i.ir. Teu.minal Divergence. 0.3S 0.0,1 Some excollont results over duplicated lines have in recent years been obtained with the Wye levels used in the engiiiL^ering branch U, S. A. The methods adopted were praetifiilly those of precision luveiliug. As an example of these I extract the following : — Section. Siuu.\ City to Fort Uaiidrtl Fort Randall to Pierre, Dak. Length in Mii.es. Max. Diver- i OENCE OK TWO OllSKRVKltS IX FEET. 17tt .()S2 190 .lud Terminal Diver(;exce IN Feet. .000 .154 The best livelling has however undoubtedly been done in Switzerland. The tirld rules there adopted are as follows : — 1. 'I'lie level inj: to be executed by equal rights whtMievcr inissible ; thediller- eiK (.■ Iii'tueeii the length of back ami fore siulit- never tn exceed ten metres, 2. The length of -ight is as a ruli' to be limited iis nndei' : — (.0 L'pun railiuails with gradieiils 1 in 10(1, to 100 metre-. (/') " " " steep gru'lient- 50. to 100 metre-. (() •' highroads in the plains :!(),to (ill metre-. (