D E S

258

D E S

construct the horizontal trace, t, of the tangent of theintersection in the point corresponding to p. This tangentis the intersection of the two planes which touch the conicsurfaces in that point ; its trace, therefore, will be in thecoincidence of the horizontal traces with such two tangentplanes. And isag'p is the projection of the line of contactof the plane which touches the first surface, the trace of suchfirst plane will be the tangent to the curve CGDu', in thepoint g' : let then g't v be that tangent. So likewise bh' pis the horizontal projection of the line of contact of the planethat touches the second surface ; and as the horizontal traceof the second tangent plane will he the tangent, in thepoint h', of the curve ehfh', let h't u be such tangent.The two tangents g' v, h' u, will intersect each other ina point, t, from which if the line tp be drawn, we shallhave the tangent in the point p, as required.

“ By proceeding in like manner with the other points Q, R, s,we shall find, first, that the tangent in Q must pass throughthe point of coincidence of the tangents in g' and H; secondly,that the tangent in r must pass through the coincidence ofthose in h and G ; and, thirdly, that the tangent in s mustpass through the coincidence of those in G and h'.

Tangents of the vertical projection are attended with nodifficulty, when those of the horizontal projection are oncedetermined ; for by projecting the horizontal traces of thetangents of the intersection, we have the points throughwhich they must pass.

“ Fifth Question. —To construct the intersection of a conicsurface, of any base, with that of a sphere.

“ We shall here suppose the two surfaces to be concentric,that is to say, the apex of the cone placed in the centre ofthe sphere, because we shall have occasion for such a dis-position in the following question.

“ Solution.—Figure 31. Let a, a , be the projections of thecommon centre of the two surfaces ; bcde, the given hori-zontal trace of the conic surface ; a m , the radius of thesphere; and the circle Ifg'm, the vertical projection ofthe sphere. Suppose from the centre common to the twosurfaces, a series of planes, which may likewise be conceivedto be all perpendicular to one of the two planes of projection :in the Figure 31, we have supposed them to be vertical.Each of such planes will cut the conic surface in a systemof right lines, and the surface of the sphere in the* circum-ference of one of its great circles ; and for each plane, thecoincidence of these lines with the circumference of the circlewill determine the points of the intersection required; draw,therefore, from the point a as many indefinite right lines,CAE, as you please, and they will be the horizontal projec-tions of so many vertical planes in the series, and, at thesame time, those of the lines according to which these planescut the two surfaces. Each right line cae, will intersectthe horizontal trace bcde of the conic surface in points, C, E,that will be the* horizontal traces of the sections made in thissurface by the corresponding plane ; and if, after havingprojected the points C, E, upon l m, in c, e, the lines a c, ae,be drawn, they will give the vertical projections of the samesections.

“ It now remains to discover the points of intersection ofthese sections with those of the sphere, upon the same plane.

“ For this purpose, having drawn through the point a, theright line gaf, parallel to l m, suppose the vertical planedrawn through c e, to turn about the vertical line raised fromthe point a, and projected in a a, as upon a hinge, till itbecome parallel to the vertical plane of projection, and that italso draw with it the sections it has made in the two surfaces.In this movement, the points C, e, will describe around thepoint a, as a centre, the arcs of circles cg,ef, and will fall

in U’l’

of th<*

ne' f

in, in G, f ; by projecting the latter points upon L /the right lines a g, a f will be the vertical projection®sections made in the conic surface, considered 'Ivente^position they have assumed in consequence of the to , er e,of the plane. The section made in the surface of theconsidered also in its new position, will have the cl | fare,ence If' g'm as a vertical projection. The'points , ‘ ! s <?y,of coincidence, g',f , of this circumference with the 1 t ;oiiaf, will be the projections of the points of in j a „ e ..required, considered also in the new position of the P ;^y ie ir“ Now, to obtain the projections of the same points, ^ yenatural position, the vertical plane must be supp 09 ® itsreturned to its original situation. In this movem en > j 0 edpoints, and consequently those of the intersection c t ]iein it, will describe the arcs of horizontal circles aio v(J j«vertical line raised from the point a as an axis, nv dra" * 1 '?tical projections will be horizontal lines. Then c>y , . £ i,e/through the points f g', the horizontal lines / ”^ n ters eC ”will contain the vertical projections of the points o . t ir«tion : but these projections must also be upon the 1 “ c0 n >cl 'right lines ac, ae , and will be found in the points o y. It.denee, ih, of the latter with the horizontal lines cgl rThus the curve khn i, drawn through all the P 01 ^ cstructed in the same manner for any other line, h e81 j r ,,d. .will be the vertical projection of the intersection ie< I g |i»l“By projecting the points, i, h, upon C E, in J> 0 f tb ahave the horizontal projections of the same P 011 ^’ a ll t^ eintersection ; and the curve k h n j drawn throu D „iifpoints, j, h, constructed in the same manner ^ 0 f tl> 8line besides C e, will be the horizontal projectiointersection. . ti j f 0 '

“ To find the tangent to the point J of the h° rl / e co/ e ?,jection, the horizontal trace p of the tangent to 'fl' 1 ’ponding point of the intersection must be consti uc pk 1 /trace must be in the coincidence of the traces otangent to the two surfaces, in the point of the t j ia t Jcorresponding to the point j. Here it is °° vl ° v j e c 15drawing c P through the point c, tangent to the cu 1 c oii

tangent to

su 1 '

we shall have the trace of the plane — .j, e _surface. And for that of the plane tangent to y. lS h e ‘of the sphere, the operation is similar to w lJ ' jj-aW

surfaces, viz- J fthe circle a

fiS

|t )U

w _ __ „ ^ -- -- . *' l'0"

described in the cases of revolving surfaces, t n, P y0

9’ o through the nnim

duced to the ri*ht * n ? e!lt to tile cuu ^ - rin „upon c e fill ? * Ime L 111 afterwards me aeai>\ ^

’ A to °> an d drawing the line o e, thro"”\ jje „

The two traces cP f Q l>tJ

through which, 1

vet"

point o, perpendicular to C e.will cross each other in a point, p,

drawn, it will be the tangent to the point J. . 0 p the

“ Hence we see that the tangent to the point ^ ^ p t0 ^etical projection of the intersection will be °h‘ aI ^ e ( j r a»vi ! ’£

tfiCi

con« en mg

ttght line ip t ^i!;^i 0n - Luili P, and afterwar

it would bp ™ ‘ n< ^ ‘ 1(3 C0I1 ic surface were not

through their t /*' es8ar T to conceive a right hue ^ jt( jiii;

planes that shn id Centres ’ an d to choose a series o ^- Sh ° uld P a " . through such line. Each ol „i

right Ji,,e f’ o a "

planes would cut the conic surface inthe sphere in one of its great circles,instance ; which would give an equally

as

tl* ;

simple

coi

ver

nst)'

tic* 11

jtin?

Distance : win,.,. .

b «t then it would -

^projection mn- ,. e a d y antageous to place the .

centres, in order 'tl^t ^ rigilt line drawn thrott f c uh"^plane to render it ‘ l ’ If moveni(; nt given to ear .

the two centre, ! paraliel to the vertical plane of

‘heir projections ^b® ma ', n - motionle8S l 80

9 tuis means the cow>ti uc

of*

simplified.

“ Sixth Question.-

-To construct the

devel 0 !’

lit

r