D E S

251

D E S

ti 0ft3 Qs ® t0 be given. Let a b, a b, be the two given projec-8 '' v av° k 6 r ’obt line, to which the generating line shouldCeiy e p Parallel; and c, c, those of the given point. Con-be j n f 0ltl this point a parallel to the generating line, it willtli e , he tangent plane required, and the points where it cutsPlane / 3 Projection will be upon the traces of the tangent

4 n i ‘ f hen if from the point c, c d be drawn parallel tothe^ ** * ro . m tbe point c, c d parallel to a b, we shall have

ti U° Projections of this right line ; then, having producedto*/ Ineet L m in the point d, if the point d be projectedi.P° n C d, the point d will be the coincidence of this°f t) le ne w ith the horizontal plane, and consequently a pointthe / Ce of the tangent plane. Now, the horizontal tracetan gent plane ought to be tangent to the curve*11 th e ’ 'berefore, if from the point d we draw to such curveVh 0n . a . n S en ts possible, as de, df, &c. we shall have thePass thr traces °1' a H t b e tangent planes that can possibly°f c 0ri j ° u gh the given point. By drawing from the points&c. to ab, the indefinite parallels eg, fh,,l ig ]j “all obtain the horizontal projections of the generat-S| )vf a es ’ wherein the different planes touch the cylindricfh, &c r astl y> we s ball have the vertical projections eg,Pfojeeti' 0 these generating lines, or lines of contact, by/ gj c S the points e, f, &c. upon the vertical plane, to'aes p ' by drawing from these latter points indefinite

bpon t/ to a b. As to the traces of the tangent plane

plane, they will be found in the working

Question. —From an arbitrary point, to draw a, “Tl le t° a given conic surface.

ghat 0 p so ‘Ution of this question differing but little from

5 ' v he re G Preceding one, we shall only refer to the Figure

/uce . 1 l e curve egfh, is the given trace of the conic

j c , thos ’ a ’ are the given' projections of the apex; andthe given point, through which the tangent“ n. U8 t Pass.

61 **W/l q .

to a ^~ Ues(t °n .—From a given line, to draw a tangentIk '* revolving surface.

8 , 8 Urf“ -Fiaure 9K Snnn,

-Figure 25. Suppose the axis of the revolv-

i, cf ion c t( ? .^ e Perpendicular to one of the two planes ofSolmio/hteh will not weaken the general application oft|/e posits eC! j use We a l' va y s retain the power of disposing

(Jthe pQ 1 /.’ been

a '. s ful e \ / n °f these planes so as to make them conform tog e ' S th e ’ et A be the given horizontal projection of the/' !| ' : Uh,, r SU| face ; a a', its vertical projection ; a pi a’, thep/ctionj q'p" ' e °1 *be surface ; and b c, b c, the two givenfYoin , he l‘" e along which the tangent plane shouldd; s ’ ; t . y ie point A drop the perpendicular A D uponjts/ee ^ 1 be t,le horizontal projection of the shortest‘‘\' e Point* 5611 t *' e ax ' s an< ^ S iv « n right line ; and pro-8iv e , w, R “ t0 upon b c.

Ship'. 1 right lj *° Se t *' e tangent plane to be drawn, and theho/R it s a* 6 to turn about the axis of revolution, withoutttu. /till lst: >nce from it, or its inclination upon theVm-,? e :.! i j' awi,, g with it the tangent plane, so that

| S'/ent, touch the surface : in consequence of such

; btti/ ' v ith t] 1S ev *dtint, that the point 6f contact of the^''u/.Phine 16 i?' ane will’change its position ; but since theMe): Com Uniformly preserves the same inclination, the

l).t H ii.!ii Ubl Unll ....i • « « i • .i n

Mi

N

Will i,/ "fill not change its altitude above the surface,■■i| Ct!,| tr e e . lri the circumference of a horizontal circle,

1 I)' • 6 111 l ' ie ax ' 3. The given right line, also,

roti/'* Sutn/* 8 - movem ent a second revolving surface|! Stem . « axis, to which the tangent plane will itself

Position.

I . ^htuct S f 11 i ' ane through the axis, and through the\V the tangent plane with the first surface, it

will cut the generating line in a point wherein the sametangent plane will come in contact with the second ; forbesides the generating line, along which it passes in thispoint, it also passes along the tangent of the horizontal circleto the same point, since it likewise passes along the tangentof the horizontal circle to the point of contact with the firstsurface, and therefore, front the known property of revolvingsurfaces, these two tangents are parallel.

“ As we wish to resolve the question by means of thesecond re* living surface, it becomes necessary to constructthe curve according to which it would be cut by a planedrawn from the axis : here we will suppose such plane to beparallel to the vertical plane of projection, and consequentlyprojected on the horizontal plane in a right line, af, parallelto L M.

“ Take upon the given right line, any point whatever,whose projections are E, e, and seek for the point wherein itwould meet the plane of the seciion it its movement. Herethe point will describe around the axis of revolution, the arcof a horizontal circle, whose horizontal projection will beobtained, by describing from the point a, as a centre, and atthe interval a e, the arc e f, till it meet the right line af,somewhere in a point, as F ; the vertical projection of the samearc may be had by drawing from the point e, the indefinitehorizontal line ef. The point F, then will he the horizontalprojection of the coincidence of the describing point with theplane of the section ; therefore, by projecting the point f tof, upon ef, the point f will be the vertical projection of thiscoincidence, and consequently a point of the section. Byrepeating the same operation for as many other points as maybe wanted, taken on the given right line, we shall have somany points, g,fr,n, through which the curve requiredmust pass.

“ Next imagine the given right line, and the tangent plane,by their simultaneous rotation about the axis, to have arrivedat a position wherein the tangent plane would be perpendi-cular to the vertical plane of projection. Here the projec-tion on this plane would be a right line, tangent at the sametime to both the curves a'ipa,grnf; and ,if all the com-mon tangents, as g i, np, be drawn to these two curves, weshall obtain the projections of all the tangent planes requiredby the question, considered in the position they haveassumed, when in the course of the rotation they have suc-cessively become perpendicular to the vertical plane. Thepoints of contact, i,p, of these tangents with the generatingline of the first surface will determine the height of those ofeach surface with all the tangent planes : therefore, if fromthese points the indefinite horizontal lines i t, p s, be drawn,they will contain the vertical projections of the points of con-tact of the surface with the planes : and if from the point A,as a centre, and with the radii respectively equal to i t, p s,the arcs of a circle, I k, p Q, be drawn, such arcs will containthe horizontal projections of the same points. To completethe discovery, it only remains to determine upon whatmeridians of the revolving surface they ought to be found, towhich end the points of contact, g, n, will be subservient.

“ For this purpose, project the points g, n, upon a G, toG, N ; from the point A, as a centre, with the intervals suc-cessively equal to A g and A N, describe the arcs GH, n o, tillthey intersect the right line B C in the points h, O ; these arcswill express the quantity of the rotation, which for eachtangent plane, the right line passing through its contacts withthe two surfaces has been obliged to make, in order to trans-port itself into the vertical plane parallel to that of projection.We have, therefore, the horizontal projections ol the sameright lines, considered in their natural positions, by drawingfrom the point A the lines A h, A o ; and the points, K, q, where