DES

259

DES

Slop'd’ an ^ ^ ase ’ an< ^ to re P resent upon the surface

ution.-

at

and,

e Mon ,1

“ &)/„,■ T a section, whose two projections are given.

* S ta ker 1 Mti0W —Suppose the surface of a sphere, whose radiusc °Ue, an Pleasure, whose centre is placed in the apex of thetio ns of I, Cons truct, as in the preceding question, the projec-t'll a p 16 ‘ ntei 'sections of the two surfaces. This done, ittio n (jjj r evident, that all the points of the spherical intersec-"i.-ie, u ° at the same distance from the apex, they must like-'[^apex n t *' e sur l' ace developed, be at an equal distance froman, l consequently upon an arc ofa circle described from“Xim as a centre, with a radius equal to that of the sphere.1 le s Urf. S ’ su PP os nig the point R (Figure 33) to be the apex of!;5 ia aem, e ,, d . eVel °P e A if from this point, as a centre, with aj^ele to a m (Figure 31) the arc, s x u, of an indefinite

a* sp] le . esc ribed, it is upon this arc that all the points ofth' 1 "'ll Intersection will fall, so that the points of suchsp), . respectively equal to the corresponding points of; ‘vi n „ . 'I)* 1 intersection. It therefore now remains, (aftertriple a f i 611 at pleasure a point by way of origin, as, forj 1 W e o„, le one projected in n, n (Figure 31), and a point s0 ( li:v fc i 0 to correspond with it on the surface developed)"tl to ^ P ^ le different arcs of the spherical intersection,# r C, f r Sure them successively upon the arc of the circled^ee f, S ’ * u certain points t. To do which, the sphericalskived np ‘louhle curvature, it must be successively%

to.

‘ a tl° l * tS tvvo curvatures, without, however, altering’itsPl'e 16 j^ln'vlng manner.

k*tni pj Pnerieal intersection being projected on the liori-

>4

N jk h, (Figure 31) it may be consideredi surfact

N J K h : this surface

t Ut)

l -° u l ( l be°K l * le su rl'ace of a vertical cylinder, whose base

W,

j'^cted j n ”p. K 11 • lit ' 3 surface may then be developed as

(l^ribed *' lr J vre 27 and the spherical intersection may be

hy developing the arc n j (Figure 31) in n' j'

, Pehrf an d ■ ‘ ' .

it" j,, . le ub

carryino- the vertical line i' * (Figure 31)„ —arly to n 1 n' ( Figure 32) from J' to The curve”'U bu T’ paS8i, tg through all the points, l", thus determined,spherical intersection, freed from its honzontalw 'thout having changed its length. The tangentifWe ^ J " of this curve will be obtained by carrying JP) measuring it upon n n (Figure 32) from J toW a ' vin 8 the right line j"p'. , „ „ „ .

Ht shall develop the curve n" j' K H n- , in

JSute , 0ld It on the arc S x u {Figure 33) : for example,h <*ni‘ 16 arc J'' from s to x, and the point x will be onfc ler i«al • SUrface developed, the point in which that of thea k JUr e o^tersection will apply, whose projections are J, *S b av 3l< ) Therefore by drawing the right line Ri we

t S lia U P° n development of the surface, the ge -

>Uy, s , e > whose horizontal projection is AC (Figure 31).

il l ’hoir Un y Point be found upon this generating line,

ip’h tv,,' 1 > brought upon the surface developed, it will be

Nft 1 * tQ ta& t J distance {Figur e 31) of such pom

st" H t T X of tlle conic surface, and to carry it {Figure 33)

dev7 R t0 v ; and the point V will be upon the

hr Vbh "''”*, the one required. .

1 <' 1 ' lri cTu .p. s<ion - _To construct the intersection of two

ok, a<:es ’ °f any bases. i .

In makib the research in which this questionif We 1. a

ii sih... ll °n,a t ess

l of n > ‘i w e i.

tw 0 r ' ave no other intersections to consider tlian

rNg S Ji av e cir^^f ldr ' c sur I aces ) ( an d especially when theseM hit- ] Plajie CU p ar ^ ases ) will be found expedient so toJSC 1 * 1 to Z 0f Projection, as that one among them mayb'Cis tl, e e -® enei ' u ting lines of the two cylinders : byV 'Std'Y ot her 1Iltersec tion will be constructed without the' ll ‘ ot|, 80 to k, curves than those given. But when we are<e rs. the,? 6 *'. v ‘ ew the intersections of these surfaces18 no longer any advantage to be derived

from a change of the planes of projection ; it will even provemore easy to represent the objects by referring them all tothe same planes. We shall therefore suppose the generatinglines of the two surfaces to be placed indifferently as to theplanes of projection.

“ Under this idea, let x f u f', x G v g' {Figure 34) be thegiven horizontal traces of the two cylindric surfaces ; a b,a b the given projections of the right line, to which thegenerating line of the first is parallel ; CD, cd, those of theline to which the generating line of the second is parallel.Suppose a .series of planes parallel to the two generatinglines : such planes will intersect the two surfaces in rightlines ; and the coincidences of the sections made in the firstsurface with those made in the second, will determine thepoints of the intersection required,

“ Thus, after having constructed, as in Figure 15, the hori-zontal trace, A e, of a plane drawn along the first given rightline, parallel to the second, draw as many parallel right lines,f G', to this trace as you please, and consider such parallels astraces of the planes of the series. Each line, F g', will cut thetrace of the first surface in such points as f, f', and that ofthe second in such as G, g', through which draw to therespective projections of the generating lines, the parallels

f h, f' h'.G J, g' j'; and the intersecting points, i>, Q, r, s,

of these lines, will be the horizontal projections of so manypoints of the intersection of the two surfaces. By workingin a similar manner on the remainder of the lines f g', weshall obtain a succession of systems of points r, Q, it, s, andthe curve that will pass through all the points so found, willbe the horizontal projection of the intersection.

“ To obtain the vertical projection, project upon L m the

points f, f' . G, G' . in ff . g, g . and draw

through these latter points, to the projections of the respective^generating lines, the parallels f' h,f h! .... g i, g i ... . andtheir coincidence will determine the vertical projections,p, q, r, s, of the points of intersection. And by thus pro-ceeding with all the other lines f G : , we. shall obtain newpoints p, q, r, s, and the curve that would pass through such jpoints will be the vertical projection of the intersection.

“ In order to obtain the tangents of these curves to thepoints p, p, construct the horizontal trace, f' y, of the planetangent in this point to the first cylindric surface ; then thetrace, -g' y, of the plane tangent in the same point to thesecond surface ; and the right line drawn from the point p tothe point r of the coincidence of these traces, will be thetangent in P. Lastly, project Yupon L ar in y , and draw theright line p y, and it will give the tangent to the point p ofthe vertical projection.

“ Eighth Question. —To construct the intersection of tworevolving surfaces, whose axes are in the same plane. j

“ Solution. —Dispose the planes of projection in such a jmanner, that one among them shall be perpendicular to theaxis of one of the surfaces, and the other parallel to the twoaxes. Then let A, Figure 35, be the horizontal projection ofthe axis of the first surface ; a a its vertical projection ; andede the given generating line of such surface. Let a b, :parallel to Li, be the horizontal projection of the axis of thesecond surface ; a' b its vertical projection, so as that A, abe the projections of the point of coincidence of the two axes ;f g h the given generator of the second surface. Conceive aseries of spherical surfaces, whose common centre would bein the point of concourse of the two axes ; and for each ofsuch surfaces construct the projection iknop qoi the greatcircle parallel to the vertical plane of projection ; which pro-jections, being arcs of circles described from the central pointa, with radii taken arbitrarily, will cut the two generatinglines in the points hp.