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I7 nf the Constitution), ^anailari Society of (^itnl (rrigiiiccp:., INCOHl'OIUTED 1.SS7. TRANSACTIONS. N.U — 'i'\\\< Sdcioly, ii-i 11 hoily, ilvji'M not hold its'.Of rosporrjihlo for llip funis nml opinions MMed in any of its piibling and clumsy, not oidy in theory but in [iraetice. In till' first place, the slide-rail is not a rigid b . nd no matter how Well lubricated its bearings, a steel rail 24 to 'iii ft. long will spring slightly in being thrown, and instead of forming a tangent to the lead there is an arc from the heel to toe and then a shar[i elbow in the reverse direction.— (See plate.) In the second place, t' ■ wheel path, even shoidil it i imform with that intendeil, is not theoretically or practically the most perfect. A sharp change in direction, necessitating an angular nnivcment of the car truck about its center, then a piece of straight line necessitating a second movement in the oppiisite direction, and lastly a curve necessitating a third slight movement in the same diri'ction as the first. Practically, 1 believe all these movements are rendered unneces- sary by play between rail and flange, and a bodily transverse sliding on the rail ; but, h.iwever they take plae.', here i- a source of increased wear and tear and resistance. Thirdly, the had rei|nired is nnmiissaiily long, which means increased yard room and more stci^l than absolulely wanU'd ; and lastly, and by no means nnimporlant, the mathematical formulas are so complex, th.at to my Lnowledgi' no en<^incer tries to ligurc them for hims<;lf, but takes what he cm Irom the tables and leaves tho balance to the foreman's eye, il' indi'cd li'' troubles about the matter at all beyond pointing out the po>ition of llio In lul-block. Take, Hu' instain'e, the following tin- the very simplest problem of all - to find the frog distance having givi:n ih" throw of switch and frog angli', etc., copied verbatim from Trantwiin's [ kct-book : tj ■■ ., ,. , colun switch angle x cotau frog J? rotf distance' ; a (gaUL'c — throw) x - ", , " eolan switch angle r colan frog - (gauge — throw i sin frog angle -in switch angle. It will be noticed that the denominators of tlie fractions preclude the u«o of logarithmic functions. The writer a'^ked one very p.iinstaking .ind competent engineer of whom he timnd one liay suiicrintendinj; the long experiet ay supcri laying iliiwii (if II (luiil)lr tliii '.v switcli witli sniiipwlmt niiusual clcnii'iiti', luiw lio (il)taimil llic jjositioiin nf IVogs, etc., ho replied tliat lu' f;ot tliem all by 8Ciilu Iroiii » [jot on a lar;;!! soaio— !.M-rtpliiealIy, in fiiot, whieli is tlie way 'I'raiitwini! i('Cotiiiricii(l,s. Now, tl-n aulliiir is itir fniiii l>i iii;; (ijiliiMd h. urajiliie nietlidds, and is e^pec,■ialiy I'dimI of tlum in hi- own prattiei' ; but in tlie ease of complicoted .switch atid turnout problems, the LTnpliic nii'tl.nd (( xi'opt as a elicek on final results when it is invaluable) is extremely ti-dious, anil ncee.s.silaicw a l»ri;e amount of trial and error work, ]ilinty of p.iper and talih,' nmni and lota of time. A man e.iniiot have all these lhin;L,'» with him in the tield, and more time is wastec! in fjoing to hi.s oflioo porhaps IJO or K) miles ami eomins l"'ck with the resnits. For above mentioned reasons, the writer hmj; airo eanio to the cfmclusion that the ler.d oi the turnout slinnld b.^ eircular from the heel of switch to the point of the frog, tiic slide mil being j)!irt of the circular curve instead of external to it, and being bent aromid stop spikes driven into tin; ties to Ibis cireular fortii. So far, so 'jood ; and he believes thiit this is not only his own pnictiee but that of nniny otl;crs. Now to find the frog distance. Let A, fig. 2, be the troir and a its angle. Produce its lino to meet the opposite rail at I', then AP.C = ", Let g represent the gauge of track AP = g. cosee a or —M— sin (/ but A.P. is the subtangent of the circular arc AIJ, and the angle being small the arc and chord are for practical purpo.ses equal to one another, and al,-o t(» the sum of tin; subt.ingcnts = 2 AP r:= frog dist« •J „ Frog distance, F = "' " ... (]-\ sni " ^ ' Now th(! number of the frog is the reciprocal of the chord of the angle, or in small angles such as generally occur on railways prac. tically of its sine. Thus a No. 10 frog has a chord or sine of ,',,. Hence approximately the frog distance F^ 2Ng. (2) g being taken as 4.7 feet a No. 10 frog will then have a frog distance of 04 feet. \V(^ shall s.u! further on how well this agrees with another formula obtained in a different way. !•' in this and subseiiucnt formnhisand ecprntions mean^ the distance nf frog point from the heel of the switch instead of from the toe, as in the pocket-book tables we have quoted. This last distance we shall refer to as the lead, and denote by the letter L. The length of slide rail we will <'all S. g is the itango usually 4.7 ft. T is the • rnw of .switch usually T) inches, which for simplicity ami with amp' .curacy we may consider a> 4 ft. Now to obtain the icnoth of slide rail H. Take the dianram in Fig. 3, a simple turnont with a I in 10 frog. F„r such a sniiUl are as 94 ft. we may consider the curve as being a parabola and that the offsets from the tangent are proportional to the sijuires of the distances from the heel of switch or pnint of cnrviitnre Now at the toe we have an offset by hypnthe.;,- o| T np . | ft. Vt the frog wo have the siaugc or 4.7 ft. Hence r^ = T . nnd S — F v/ -' (^) For all onlinary railway eases T and arc constants and V ^ ::: L".! Hence 8 = '29 P (4) or in Fig. ;j s •20 X 94 27.3 feel. Fig.l Fig. 6 ■^yS wAm^ Fig 10 l\ = GG.fi Li = 40.(uuarly) Tliia Irts! problem cau ofcuurse be readily solved bv ii drawiucr. but nut wiilijut oonsidurablo conatruotion vork and laboiu-. H. onco S =r -2!) P (4) or in V\ir. ;j S —. -2!) x 94 i_ 27.3 Ci.,'! 2 2Ni 9.4 sin " or t).4N 'I'licw tliioe c<|iiation,s : HID u P=: .2!) P iinil L = F— H (5) will w)lvi! :i|| tho simpliT prolilems of sinj;io throw tiininut, but in tlio (limble ami troblo throw souicthiiii; iiKirc is wiiiitcd, h was eases like ilin. 4 ami ."> which .-ct tlio writor thinkiii;;; on the follnwinj,' lines. I'otiirnint; to fi-;. li again, the ott'set at the end of a 100 feet chord of a 1° curv(? is 0.87 ft., and ineronsos proportionally to ihn degree of ourvatuie. Ilcncc for a curve of [)° it will bo .87 D and aa above for any other distance (within moderate limits) from the point of cur- vature, it will bo for N foot .87 D . (JL)' The total deflection'iit the end of this distance will be Then for fi''. 3 we have _ .; 87 n — „ r„\ 100 °' " — n W and X D 100 " CO SimplifyinK Qt) and (4) F ^D = 10'!£?.K „„d FD ^ 100 a .87 F "D -^ 100 " F therefore 100 n F = lOOOOj; ;.87 Subftitutini; in (l>) F = 100 n .87 n ino fv Tnfn D loo 115 D (6) .87 (7). and Substituting for— „ and -—the same values as before, 1 in 10 or 5°.75 — and for g— 4.7 ft , we get : P = 94 ft. as before D = 6°.12or1B°07' 8 = 27 ft. L "= ()7 ftT Cemparin- with Trautwini.'s tables, we find for a 26 ft. slide rail, a lead of 72. 7 or F --- !)0.7. and R--S77or D:=6° 30' We have thorefore saved fi ft. of load or 12 ft. of steel, and got a slightly easier curve on it. Lower down in (lie same table we will Qiul that with a Hi ft. slide rail, we will have a lead of 07.8 and a curvatuio of 6'' 15' practically the same as above, but with an angle at the heel of 1" 30' whieh is rather abrujit for high speo.Is :iiid eertainly not as desirable as a con- tinuous curve. Let us now tike up fig. 4, a tl.ree throw switch with two 1 in 10 frogs, on the main line. Wt- wish to determine the longitudinal posi- tion of till' third frog. We have given the cu.vature I):-(i°.12 the oftsot for whieh for 10(1 ft. or 0~=.87 D = r. :! ft. (fiom the tabhs) . Sine; the tl:ird frog is eviileutly in the centre of the gauge, we have the total nffaet— i g or 2.3.'). hence the ei(ualion (looj ' • 5.3 .^ 2.35 Fi' = 4434 F, = CO.fi Li = 40 (nearly) Tliis hist, problem can of course be reudilv solved bv a drawini' not wulijut considerable eonstruotiou work aud labour. but Tlio fi)r,ii\vlii^' fij;. ,"i is iidt ((uiir Mil siiii|iif, liiil it is priibiiWy a bctler armnsrciucnt ln'oiiusc no twu froj;scoiuc opposite ouo othiT, ami tliorc i» D) point on tiio main lini' unfiotcetcd bj- a i;«'ii(l rail as tliorc is in fi,'. 4. Tlio fioi; in north rail, fiu'. 4, boini; 1 in 8 from (>.|ualion (2), wc at t : F = 2 Nd r^ 2 8 X 4.7 := 75.2 It. from ^4; S = 29 F — 75.2 .29 = 2,1.8, say 22 ft. (5) L = K_S == 75.2 X 22 = 53.2 (7) 1) = S7 1^" = •"" ''^{""^' - !).°49or9° .30' nearly. NoWj'to dotcruiiuf ilie position of tlie third fiojj, which is not now in tlie ccntro of track. Tliu tan^i iiti:d off-*ot fw the .-outh curve is as iil'ovc, 5.3, for ll (■ noilh 8.:!. If F (jo tho fio<; dislan. ? on tbi> north line, Fit; will be tliiit on tiie Miiitli, tiie >« itch being C ft. Ionizer, Thi.^ we may s.iy, before ;,'oinj; fiir lior. involves no difficulty in prac- tice, the spikes on tho north side being rlriven ft., or 3 tie."!, nearer the tijc than nn the south. Now we have for the distanee of tho frog from the north rail the expression 5.3 \^ 100 / " from the south rail the expression (i,.,,|l x 8.3 But the sum of these two is the fzaiige. Whence we get tho equation : — Simplifying— (F' + 12 F + 30) 5.3 + 8.3 1-^ = 47000 13.6 f ' + 63.6 F = 47000—191 = 46809 F>+ 4.68 F + (2.34)' = 3442 + 5.5 = 3447.5 F+ 2.34 = 58.71 F = 56.4 F + 6 = G2.4 L = 34.4 This determines the Idntiitudinal position — substituting the above values^in the expressions alucs. / F + 6 \^ / F V (' TcTTJ " ■*•■' ''"'' 1 100,^ ^••'~''' ^"' "'" '■ Distance from XorthiRail 2.06 '• " South '■ 2.64 4.70 The angle of the frog will be the sum of the deflections of the two turnouts. "- Wo'''-' '-^ "'■'-' 15J6F+36.6 leo 9.°16or 9"10' A No. 6 frog, jiinclied ;i little with tl!i> crow or mil bender and with the point set back a tiiflu from tlie tlieoieticiil as obtained above will fit the place quite well in jiracliee; but if we wish to be very exact about it, tlic following; analysis ol'the problem in Fig. 6 a double throw both on same side of main line will suggest a means of c deulating. This is not a couimi u combin;itii n, but ii aetually,or rather one almost identical with it, occurred in the wriier's pi:ietiee. One turnout was for a main siding and the other f.r a V, and in order to save room and switch frames iiiid uiinece.-sary eonijiiieation it was decided to start both from the same headblock. The main turnout was alnady laid with a 67 it. lead and a 1 in 10 frog, as in Fig 3. We had on hand a No. 6 and a No. 10 frog, which we desired to use. Putting the 1 in 6 first in the main line we get from eq'iation (2) F = 2g X N = 9.4 X 6 = 56.4, F 23.8 ", then fore l> - 5G.U ^•^.W or ft- ill siirli all .liiin.' angle as 9'-' 32' llw siiio and cliord commence titdiffoi' ii|>|)iipiaUy— vv m ly take tluMiioic cxftft \ahu- by (1) : IV 'J.4 Hn " ' sin U" 32' T is in tliis caw 10 inclics or .83 ft. Subslltutini: iu (3) s;- V \/ J- F l» 1(10 = lC°.7a or !(;>■ 44' We liiivf still to iiltiiin tliu jiosition oi" the 1 in 1 frog— between tlie two LuiMout rails. I.tt .i- be the nr.kn.nvn distance from the point of the last one toii.siilrrcd (the 1 iu (I), and 1> thv curvature per 100 ft. over tliin piece x. mi X D 100 *''"^''""' detlcctioii of till' South tiinioul between the two fro^s, and the total dcflectiuii Iroui lua line tn point of 1 in 10 frog will be '-- ^ O*-' 32'.. On the north liinioiit the dettcction tj same point will ho that for a dh-ftaiice 27 + 33 i ;<: = x + 00, and will be represenUd by the exprcsKiou -".^j^ - x (!".12. But by hypo- thesis the difference hetwton these is the anjrle of the frog ov 5'' 14' hence the equation : ) ^ , en . 5.«'73 lOo *■'■ ^'^ ''-'2 Too xD+ !)53 — 0.12 a; — 307 = 573 xD=6.12.T— 13 „ ^12 a— 13 X Again we have for the lateral position of the frog from south rail of main line : — on the north turnout (— yy— )'x 5.3 on the south " x sin 9" 32' + 87 D / ^' ] (60 + xVx 5.3 i,.rfl .87 Dx^ ciiuatiDL' - ■Iti.'ifi r + ' '^ lOtiOO 10000 (fiO + x.y X 5.3 = 1656 X + .87 D.t,' 1H080 + 630 X + 5.3 x' = lri56 x + .87 D x' substituting the value of D in terms of k above. 6.1 X - 13 5.3 x' + 636 X + 19080 = .87 .»■ ^ x + 1656 X - 5.3 ,r. ^- 11.3 X + 1656 .X 1009 a; = 19080 .1- =18.9. Substituting in the equation of D in tornis of x, I) = fi.lx-l;i 115.3-13 18.!t •5". 41 or 5= 25' Substituting in the expression for the offset or lateral position of the frog point, we have : = (TOO ) •'••'=•"»'>■ •'i.3 - 3.3 ft. Fig. 7 is a eoiiibiiiatiou of the arrangoiuei t> in Fiys. 5 and 6, and is an example of a 4 throw switch, something i„ t often seen in actual prac- tice. For reasons explained fnrtl.er on. it would nm ho advisable to break up the sharp curve foiming the lead of south lurnont by insertina; another fro.y;. A 1 in o has tlicivloic been instituted. Making F = 75.2, I) = 9" 30', S = 31.6, L = 43 6 The only elemenl left not solved as i'l the preredini; is the positioi" of frog between the extreme north and south turnouts and itfl angle. 5 Both trnrks liavr mnv :i curvuiiiiv of !» -.'iO. = 8.:t, hut ono ourvcfUrts 10 ft. l...lii.i,| tli.' .-tlnr. W',- Uw ilim th,. Kiiintioii : — 2 F'+i'O V , 10n=.- 4"00't .^„_, I'-" > 10 F + (5)^ » L>8:il - r,0 ^ 25 =, 280(1. F + ■> = 5;i ^,. F=IS L.2ti i)i>tanoo from-outli rail ,in,I main lin,' ns almvo F- X 8 ;) l.HMm = '•" ''•"'■■' 11" For the anj;li,' nnVn;,' we liav.> : 2 F ^ 10 " = —,7;; X !'•■"• -- 10.07 or 10° 04' It is quite possible u. add a Ulth irack making a 5 tlirow 8witrl, • but a« .t necosaitato. 4 extra frog, and 8 ,.„„rd r,,ils, tl,c extra con.plexity and .o,f v,ll generally ,M,tw,.i.I, tl,. advanta,..., of convenienco in operation. borne of tl,e writer'Mnathcn.atieai contemp„rnr=es, wlm discs. .d the trouH.tioneurve, will, l,e fe..ls.s,.>c. to dissatisfied with some oftlie above methodB, because they an not preeise. For the ..„ke of sinmiVity ]„ resulting; lorm.las he has treato.l the curve of the turnout at one moment as a eneuiar are .and at the next as a parabolic. IJe has assumed the sum of the subtan.^-eots as beini,' oqnal to eh.rd and 'the CO sine of a considerabl,. an.le : !vi„. .,„i,j, ,li, ,„,„,,, „ „,„„, i^- tha a difference of ^ fow ineh.s ,n the had on either side will make no differ, nee whale, r in the piaetie.d utility of the results; and a. Irojis are made m a forge ami machine shop, and not by a mathoma tical instrument maker, the errors in th,.,,, will „f„,„ ainount to „,ore th;.n those du,. to the want of pn ..jsi,,,, in the fonoMlas. Vud further wheth.T the curve be ei,vula,-, pariholic or sou.elhin- a little 'different fro.,, either niatrcrs uot ,me particle, provided the radius of curvature is nearly coust.nt and there are no sh.,rp a„.,les o,- bivaks in it The methods and tbrmulas giv,., al.nv,. have be,.,, np-atedly tested in pri.ctice, an,l found to yiel,l good ivsults in every ease. On the other hai„l, ,s„me of the praeiical men will 'say that it is use less to spend tunc an,l figures on sueh w„rk, and that a fb.emnu with .'i good eye will run them o;;t by I,ims,.lf.i„st as truly as can bo done by the engM.ee.. Tl.e writer has been a go„d d,.al with tra,.k n,en, and has repeatedly olbeiat.,! ast„,.oman hin,.seif, and believes ho has .,st,.„e an eye as „,ost. lie has found that, it left to ll, msoiv.s tluy will make even ,h,. host of then,, the most surprising errors, not of inch,. b,.t nf feet m t'„. simplest p,-„bl,„,,, such us in Fig. 'j. Tl,c truck layer who is judge,, m.inly by ,,ua,.,i.y ,.;-v.rk be d„es, wili alnios; invaiiably «ct his l..a.ls t„o ^hori ; :u„l ,1„. .s,.eti„n.foreman, who is judged by the ,|ual,ty, :„„1 who m ,.s the fa.linur, of his p,cde,.ess,u.'H work, ,s ,.m„st invariably impivss..,! with tl,.. iienssi.y lor very l„n,. leads, and acts. ee„„l„„ly. As . n„.t„.r „na,.,, a l,.,! , xtre„„'v h,„: necessitates a. slaip ki„k s„„,e«lK.,., ju.t as „„ ..Mr,.m,.|v sl„.,"t o,,:. do(^ andisonW a triH,. less objectioMabl. ,„d uns,gl,„y. Sncl, problems as hav b,,.,, m,1vc,1 in tl,c. prec.lin,. with the u.scof only one unk,„Mvn ,|uu,u,ty may ., ,„.,.ally 1,.. work,.d „„t on the ground by any tolerabbMutelligenl man by stakes and tape m.-anirement, in .somesu,..^. way as frnutwine sugg.^ts, but this is n,.,rcly a graphi,.,-,l solution on a full s,z,>d drawing. On a i-a,! where trains are p.assing a,„l .so.nc times Stan, ,n,.r every l..w „,in„es, it will not b,= found a very econom- ,cal metho,!, especially >vl„.n h„lf u ,lo.en or more „,enare waiti,,.. to -^o to«.,,-k;andm:„,y ease it is a looger , operation than the algebniie solutions giver, above. The proble.ns above, in which we hav; u^ed two unknowns, are almost ' „l,.terinin.te on the ..round, ..xcept by the nietho,! „ succssiv,- trials an.l approximations, which uL is not economical ot ti,„e. ''''■'■" lll''|"'^''i I' IVnjis iiimI tor nlMvit.'h, .ni'l i.liiin-t n„y fufi' mail ii!:i\ lie ini-t.il t.i muko ;in I'liir u lirh' us n.'i il h,.. Tlic wIkiIc Hll.j.'.l is piwi>fly ,iIllM)rillO>t illVolvillj; lln! iMIlltlOIl- pi.ii',' .IrlailM of 11 iMilwiiy, "lii.h ,iro m) ortcil ncgkctril as tu.) simple aiil w, II iiMihr.il.iiMl lo Ijc; worth t^ir att.iilh.n of 'i fir.-l clans ciujiiK-cr. I \iin-r iii-taiiunl iMic vii'^iiicci' >vlio ;y Ibi- . fnv- tli(Tc.\j,laiifttic'ii ofhisckvuiiiii; it w.itli will.' 1.1 stuily th.' i,ti,vstinn tiiid to lay the r.-siilis Imforo the tMiVniiMiiiii- Iratornilv. Split .^wittlicH HI,; (li,,plnciii|,' tlic old fasliinnod stub 8witohcH wi- liiivo lurttofori; bocii considiriiij,' for a,iy sor/icc involviii;.' lii^uli spood. The advantage is .d'coiirsii the (•ontimiity of tlir rails and tliu alisi-nco of jar arid jork. Tlio olyciions an liislur ni.-t and lliut tiny arc not ciisccptiblc of licini; wnikcil int.. ;i ami 4 throw mruoiits, licnec they will probably never displac them altogether. The split switcli differs essentially from the stiili, in havin;' a luovealil.j rail planed down to a bevel ed-o bavin- an ani,'l,! of from 1 to 2 d.grees, aii.l movi;iji; into position alm-sid.; its nei-hbonr instead of vlniirU u> its end. There is tlms an inevitable nnj.de a.s iti the stub switehes dis-eussed in the pocket oooks. As it is iuipracticafale to brinj; this bt^vudlinu' to an iK.tual feather edge, tho virtual point is a fort or so back of the actual. For convenience of cal 'ation ami nth. r practical reasons, which need not no gone into here, bn„ ,vhich may br midiTstnod from a >tudy nf the diagram of the position of the wheels of an ordinary 8wlieel'..l en'.'in.' on a curve (Fig. 8), th- writers practice is to eonsi.ler this angle as merely an elbow in the continuous turnout curve, as work.'d out in tho tirst example, Fig. :!, and th.' virtual p.,ii,t as the intersection point of tho snbtangcnts eorrespondint^ to tho angle. If, as in Fi.', 1, the frog angle be 1 in 111, tho curve (i" and the angle of Hy rail l°;iO'we shall have these subtangents I^.-t fi. each, the I'. C. will therefore be 12 ,'- ft. back if the virtual, or l;{..5 ft. from the actual p.iin' of tho fly-rail, and the inner edgi> of this last must bo straight tor llj ft., uftor w'.iob it will be bent to the curve of the turnout. Its total length must be at least sufficient to secure the necessary clearance for tho wheel llangcs, which we have heretofore taken at .4 ft. For i.n angle of l'^30' this will be 1 5 ft and for 2" 1 1 ft. The total frog distance in Fig. ,'! was 04 It. Subtend i.c Ili.f), we get 8(1.5 for tlie length of load. With these eonsideration.s borne in mind, the viirioiis problems for 2 ami .t throw turnouts may be work.^.l .just as in the case of stub switches. I have already said that .'! throw swittdu's ar.' impossible with the split switidies, but it is .piite common to lind a douhl.' turnout arranged as in Fig. 0, with ope swit.di a little ahea.l of the other. Kiic.pt that the ili-iances are Kmger, this is a preeLscly similar ease to that ill Fig. 5. Tin; writer has already allu(ied to the diagram in Fig. S. It is introduced for the .sake of demonstrating the objection to a frog in the n.iddle of a .sharp curve. The leading ilriver will inevitably strik.' the elbow, if in. Iced it .Iocs not ri.le over it ; ami ih.' trailer will vreneh the guar.l rail badly. The truck wheels traicr.-ic well enough, and as b^^lli drivers teinl (..wards the iiisi.l,' rail the point of frog is easily t.iken etirj of. Hence, while the load may be luade fairly sharp with safety, then' sh.ml.l always he a piece of straight or easy curve bey 1 the IV.ig, heiii'e (lie flatt.'tiing in Fig. 6 in the sciitli turnout between th.> 1 in (i an.l 1 in 10 frogs, in rci^ard to guard rails, (hey will be M^eii in the ol.ler roads iiia.le s.i as to bo parallel with the main rail hir .juit,. a .listane.% vith a sharp kink i;t eai'b end. Fig. ]().(. These are required to bi; made in the foig... Th,! wiHer has always bc'ii in the habit of making them as in h'ig, 10 /., le.idii,.' the stray wheel gradually into its proper line au.l then hating it ..^o immediately di.' poini .d frog i^' p:isse.l. These can be nia.le on the ground in a few minuCes with a jim crow an.l c.i.l chisel, and he is glad to sec them coming iuto general use. If, a-^ tho writer Violicvc-;, tlio inctlioils and priiicipli's I'luincintn.l iibovc Fiave iiovc licn used in connection witli tiiinout ]iroblenis oxrept by liitni-ell', he thinks the inainteiinneo of WMy enijini'er will be well repaid liy a stmly of tliem, as he has himself heen in the ineioasicd rapidity witli which he can iirrive at accurate results, and which is perhaps of more conscijuenoe in clean cut easy curves and turnouts. InaFmueh as tho equations oficn involve lar;,'e numbers, s((Mares and s()uare r ots, lie can never he far astray in plottino: the results afterwards as a (dieck on the arithmetical work. It is niuoh easier and takes less time to plot n re-ult than to iirrivc :il it frraphically; and as an evidence of its being worth while to employ some means of locatin'^ frofjs and swilches before L;oin;; to work, it is only necessary to point out one or two of the many clumsy, ill tittini^ ones which any tyl'o can detect in ala'ost any large yard as \hv resnll of baphiizard work under no system whatever. 8