DE S

256

D E S

would pass through all the points, i, A, thus determined, wouldbe that projection. By projecting the points i, A, in J, k, onthe projection, oke, on the corresponding vertical plane, weshall have the horizontal projection of the same points, andthe curve, kjp, that would pass through all the pointsso determined, would be the horizontal projection of theintersection.

“ In seeking the tangents of these two projections to thepoints J, i, it must be recollected that such tangents areprojections of the tangent to the intersection. Now, thelatter tangent being at the same time in the cutting planeand in the tangent plane to the cylindric surface, it musthave its horizontal trace in the intersection of the horizontaltraces of the two planes : the trace of the tangent plane isalso the tangent in f to the curve c*df, Therefore, bydrawing this tangent, and having produced it till it meet thetrace of the cutting plane in a point g, drawing the rightline G J, we shall have the latter line touching, at the point J,the horizontal projection of the intersection. And by pro-jecting the point g upon L M in g, and drawing the rightline g i, we shall obtain the tangent, in i, of the verticalprojection of the same curve.

“ Should it be required to construct the curve of theintersection as it is in the plane, suppose the cutting planeto turn upon its horizontal trace h g, as upon a hinge, andapplying itself to the horizontal plane. In this movement,each of the points of the section, that, by way of example,which is projected in J, will describe an arc of a circle, whoseplane will be vertical, perpendicular to H G, and whoseindefinite projection will be obtained by drawing throughthe point J a right line, R J s, perpendicular to h g ; conse-quently, when the plane is laid down, the point of the sectionwill fall somewhere -spon a point of this right line. Todiscover the distance „ such point from the hinge : as thehorizontal projection of this distance is j R, and the differenceof the heights of its extremities is the vertical line is; if j rbe measured upon lm, from s to r, the liypothenuse r i willbe such distance. Then by measuring r i upon r j, fromR to s, the point swill be one of the points of the intersectionconsidered, with its plane laid down upon the horizontalplane, and the curve s t tj v drawn through all the points sconstructed alike, will be the intersection itself.

“ To obtain the tangent of this curve to the point s, it issufficient to observe, that in the movement of the cuttingplane, the tangent does not cease to pass through the point Gof the hinge ; therefore, by drawing the right line S G, weshall have the tangent required.

“ Second Question .—To construct the intersection of aconic surface, of any given base, by a plane given inposition.

“ Solution .—Here we may suppose, what is alwayspossible, the vertical plane of projection to be placed perpen-dicular to the cutting plane.

“ Figure 29. —Let A, a', be the projections of the apex ofthe cone, or of the centre of the conic surface ; b c d e thetrace of such surface on the horizontal plane; fg the verticalprojection of the cutting plane ; and G^/its horizontal trace.Suppose from the apex of the cone a series of planes, per-pendicular to the vertical plane of projection ; the verticalprojections of such planes will be the right lines a' c, drawnfrom the projection of the apex : and their horizontal traceswill be the right lines c c, perpendicular to lm, which willcut the trace of the conic surface somewhere in the pointsC, c'. ... These planes will cut the surface in right lines,

whose vertical projections will be the lines a c .and

whose horizontal projections will be obtained by drawingfrom the point a, the lines c a, c' a _; the same planes will

fiX*

jt^alar w

likewise intersect the cutting plane in lines perpenu ^ thethe vertical plane. The projections of such lines wi , ,}

points, h . . ., of coincidence of f g, with the lines j fl g,and their horizontal projections will be obtained by a u [ar9

from the points h ___ upon L m, the indefinite perpen j.

An .... This done, the lines An .... will cut the corf ^ing lines C A , c' A .. ., in points, h, h' ...., which "'! ct joflhorizontal projections of so many of the points of in t | )r0 ughrequired; and the curve, phqh', that would 0 f tb e

all the points thus constructed, will be the projectsintersection. . t

“ To draw a tangent to this curve, from a p olD ^ jjor 1 'upon at pleasure, it is' only requisite to find upon t j 0 n i®zontal plane the trace of the tangent of the inter® ^ (jjjta point corresponding to h. This trace must be 8 l9<>of the cutting plane, and consequently on G fs jn tb®

be upon that of the plane touching the conic sur aright line, whose projection is ah : and if A B he Cj tb®till it meet the curve BCD* somewhere in a P°' t he h 0l Ttangent, c f, of this curve, in the point C, will be of« oinCzontal trace of the tangent plane. The point, F, th edence of the two traces f G, c f, will therefore otangent in the point, h, of the curve phqh'. - on c0®'

“ Should it be required to construct the interse^^sidered in its plane, we may indefinitely suppose ei 0 r(kthe cutting plane turns upon g/, as upon a hi n S e ’ uC tto apply itself to the horizontal plane, and so con jt tu rl \curve in the position it will thus assume; or t 1 yerti® 4 *upon its vertical projection fg, to apply itself to .plane ; the latter is the hypothesis we shall here a plan“ All the horizontal lines in which the a e rie9 _] aP e, ^drawn from the apex has intersected the g j$e*

which are perpendicular to f g, do not change ^ ease tothe movement of the cutting plane, nor do theyperpendicular to fg: if then indefinite perpe® „ t he!Ddrawn through all the points A, to f g; and it ^° nsU red ® fl fcorresponding horizontal linesapplied from A, to n and n', the points N,the section ; and the curve rnsn', dra~points thus constructed, will be the intersection ^

in its plane. •dent, tb®*’.w

“ From all that has been said, it appears evidraw to this curve a tangent in a point, N, taK ^ fj-o*® ce &upon it, the perpendicular N A must be dropp p r ° la 'i r o'point N upon f g; that the right line a A nius be P

iu i* -r . r efl 'drawn t ^ iroU ® oia sid er

tiii it meet l uZi thT ^ n *£ bt h'ne a ,l Mai -- pijected i a c> or tJ P"int c ; that this point must ^

drawn to such curveTn c ° ° E ; tbat th « tang f,n a point, F- and ti ! C ’ cutting the trace G /

from fin 0 nn mast he carried perpend'*'' 0 tsquired. J ’ lhe «ght line o jv will be the ta» a

conic surface >° f constr ucting the development^

‘heeffectof'heLr a ‘^ er base > ™d of tracing j' %,11

proceed to describe if rseatl °" with the cutting P hne ’'Lcti o! ‘of the conic surface ’ 3 ^ ter lav,n g spoken of the in 0 gethe apex. ° Wlth th *> ot ? a sphere having

vrcuIarZTefZll^ intersection of two conic surf^ 3 ^

“ Solution L. fc- B a * es are parallel to each othe 1 ' ^to.*Peat wTaU^T Asit would be supejff^l

method, of which tl been sa ‘<l in treating of 1 j Jj 0 oifobserve! tbaXinZ * ^ Ure was a type ; we 3 ]f t0 otwo revolving surf 6 ° aSe II0VV before us, as well as 1made in the two whatever description, 1 1®

hut we shall enter 3teS tbe horizontal plaMCS, ^ ' ^, o

which we hate n e Upo " 80m « details relative to tavS ePnate not y et h ad occasion t0 speak .