DE S

DE S

263

*he 'visn°{’ as ' Knowing the angle formed by the zenith withra y directed to the point A, he is also acquainted, ail gl e formed by the same ray with the vertical linee art j 0ve the point A : for, disregarding the convexity ofNt 0 i which is here admissible, these two angles will be?? con 0 „- ? a 'ternate-internal, and consequently equal. Thenlri th e riA;I In ° a c °nic surface, of a circular base, with its apex

osQeratj ' nt . A > its axis vertical, and forming with its axis and^Pletel^ ^ ne an an gi e equal to the angle observed, whichv ’SUal r ^ ^termines such surface, it will pass along theSoin^-ted to the point A, and, consequently along^tface d °* station : thus he will have a first curved% pro^ained, on w hich he may find the point required.le fi rst 8n a similar manner for the points B, C, as for%c s ’ he Point required will farther appear on two other

ae ' 1 ' an; Ces > w ’th circular bases, having their axes vertical,.P'Ces in __ j ...n _

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axi^ and 'i* ^ le points b, c, and each forming an angle with

* et >ith generating line equal to the angle formed by thej,° r j the re f t,e corresponding visual ray. The point sought0 j. ° re > will be at the same time upon three conic sur-sf c etertQ i na te forms and positions, and consequentlya „ mra pn intersection. It remains then only to con-H vert:° 0 , r ^i n S, to the data of the question, the horizontal

^ DrOlP/'tlATlB +V10 intm-fn/ili'Ana nf +1) ocn

8 , , Projections of the intersections of these three“a 9 a . e n by two’s ; the intersections of these projec-?'at ren5 1Ve ^ le horizontal and vertical projections of the■ its’ an( i consequently its position on the chart,

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‘eh

l . wiau ita pusuiou uu me uinui,

ijl e *ght above or below the points of observation,

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-U Will A ttUUVtJ urThj, aet crmine the quota,13 solutj- -

tin' Ver to t(, tl0n will generally produce eight points int|, ish am 6 < l uest ion ; but the observer may readily dis-^ st ation ° n S them that which coincides with the point ofa j 11)1 of t ] ( ’ He ma y a { g rg(; sight ascertain whether thethe th Stat ' on he above or below the plane that passesa i>ov e Points of observation. Suppose this point toe ct s Uch if ane cP the apices of the cones ; he may thenPo 8 ^ e heloyy. , aric h es of the intersections of the conic surfacesnfjHle p 0 j tie plane, which will reduce the number of\ ] 8t ati otl tS !° fonr. So, on the contrary, were the point

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?v e

; 8t

be eQ n PHced below the plane, the number of pointsTi ^ re< iuced by omitting such branches as were

it.

a’ he tt ,;n len among these four points, if indeed so many

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st a .. three e . asi iy recognize that whose position, relativelyn'H rel a ^i )1< ? es > i g the same with that of the point of theho t : 'J.y to the points of observation,

the ^tal p ro ?° w -; —Figure 41. Let a, b, c, represent theaa rt ; a J®°tions of three points of observation taken onby J ’ c > the vertical projections of such points, con-on the vertical lines b b, c c, beginning0 f 2 ° n tal line l h, passing through the point a, theth"’ r ®pre s ae < l u °tas of the two other points ; and let'< y, ^speetf 111 t * le angles which the visual rays, directedHi th e V - 6 P°ints a, b, c, form with the zenith.

Ve r t -I1C ^ n * te ver tical lines a a', b b\ c c, and theyH J th e *1°^ P r °j ec tions of the axes of the three cones ;

’ e \ and' 166 P°’ nts a > b, c, draw the right linesN s en ** ^ e y will form with the vertical lines, angles* b e t ° the given angles A', b', c' ; which right

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."o each ““s*""

s Hes of tv. ttle vert > ca i projection of one of the two• ^°t)e -j corresponding conic surface.of?‘pC raw * n tb e vertical projection horizontal

;'s uf ____ ^__

iJ ^ 58 ta an a8 p re ) which may be considered as the projec-(( H\ We or izontal planes, and for each of them workV , are about to describe for the one indicated

V>wi U

* 1 °inta f C j t '•be projections of the axes of the three5 9, h, that are the vertical projections of the

centres of the circles, according to which the correspondinghorizontal plane cuts the three conic surfaces ; it will alsocut the extreme sides of the cones a l, b m,e n. in such points,f, <f, h\ that the distances/ 1 f g g' hh', will be the radii ofthese circles. From the points a, b, c, taken successively ascentres, and with the radii respectively equal toff, g g’, h h’,describe circles, whose circumferences will be the horizontalprojections of the sections made in the three conic surfaces bythe same plane ee' ; these circumferences will intersect eachother, two by two, in points d d', k, k', j, j', that are the pro-jections of as many points of the three intersections of theconic surfaces, considered by two’s ; and by projecting thesepoints upon e e', in dd’, k, It iit, we shall obtain the verticalprojections of the same points of the three intersections.

“ By afterwards working in the same manner upon theother right lines e e we shall obtain from each of themnew points, D, d , k, k', j, s', in the horizontal projection, andlikewise, d, d', k, k, i, i', in the vertical one ; then pass a curve,

D P d', through all the points n, i> _and it will be the

horizontal projection of the intersection of the first conic sur-face with the second : pass another curve k p k', through allthe points K, k' .... and it will be the projection of the inter-section of the second surface with the third ; and by passing a

third curve, J p /, through all the points j, j' _, we shall

have the projection of the intersection of the third surface

with the first. All the points p-wherein these three

curves intersect each other, are the horizontal projections ofas many points answering to the question.

“ So, in the vertical projection, by passing a first curvethrough all the points d, d'.... a second through all the pointsk,k‘.... and a third through all the points i, i'.... we shallhave in such curves the vertical projections of the intersec-tions of the three surfaces, taken in pairs ; and all the points,p ... wherein such curves intersect each other, will be thevertical projections of all the points necessary to the solutionof the question.

“ The projections p, p, of an identical point will be in thesame perpendicular to l m.

“ Having discovered among the points p the one indicativeof the point of the station, the observer will be in possession ofthe horizontal projection of such station, and, consequently,of its position on the chart; then by means of the altitude ofthe corresponding point p above the right line L m, he mayobtain the elevation of the point of the station above thatof the point of observation A, and that will give him theappropriate quota.

“ In this solution we have constructed the projections ofthe three intersections of the surfaces ; but two would havebeen sufficient. Yet we would always advise the adoptionof this practice, because the projections of the two curves ofdouble curvature may intersect each other in points not cor-respondent to those of intersection ; and also, because, inorder to recognize the projections of the points of intersection,it is necessary to follow the branches of the two curves thatare upon the same face of one of the surfaces ; which requiresa laborious degree of attention, seldom, if ever, necessary inconstructing the three curves : the points wherein they allthree cut each other, are true points of intersection.

“ Fifth Question .—Under circumstances similar to thepreceding, except that the instrument is not furnished with aplumb-line, so that the angles formed with the zenith cannotbe measured, the engineer is required, without quitting thestation, to determine on the chart the situation ot the pointwhere he is, and to find the quota belonging to it, viz. itselevation above the level surface to which all the points ofthe chart refer.

“ Mode of Solution .—Having made choice of three points