--

;

DES

248

DES

CP, D Q, upon A h. The horizontal projections of the twopoints of contact will therefore be found upon the two rightlines CP, d Q. But in the retrograde movement of the per-pendicular plane, the two tangents J C K', J it D do not ceaseto pass through the respective points of contact ; and whenthis plane is returned to its primitive position, the point J isprojected anew in G, and the two tangents are projectedaccording to the right lines G k', g k. The two latter, there-fore, must each contain one of the points of contact ; and, infine, the intersections of these two right lines with therespective lines C p, DQ, will determine the horizontal pro-jections, R, s, of the two points of contact, which are foundin the same line with the point n.

“ To obtain the vertical projections of the same points, first,draw the indefinite perpendiculars r r, s s, upon L m ; then byprojecting the points K, K', to k, k', and drawing the finest?g k', from the point g, we shall have the vertical projectionsof two similar tangents. These lines, therefore, will containthe projections of the respective points of contact ; and thepoints, r, s, of their intersection with the vertical lines r r,s s, will be the projections required.

“ The horizontal and vertical projections of the two pointsof contact being found, to construct, upon the horizontal planethe traces of the two tangent planes, lines parallel to thegiven right line must be conceived to pass through each ofthe points of contact. These lines will be in the respectivetangent planes, and their horizontal and vertical projectionswill be obtained by drawing R u, s v, parallel to E f, and r u,s v, parallel to ef. On the horizontal plane, construct thetrace, t, of the given right line, and the traces, u, v, of the twolast lines ; and the lines t u, t v, will be the traces of the twotangent planes.

“Instead of supposing fresh lines to pass throughthe points of contact, we may find the traces of the twotangents G R, G s, which will answer the same purpose.As to the traces of two similar planes with the vertical plane,they may be obtained by the method already so oftenalluded to.

“This solution may he rendered much more elegant bymaking the two planes of projection pass through the centreof the sphere itself. By this mode the two projections of thesphere would be mingled in the same circle, and the produc-tions of the right lines would not be so long. We have onlyseparated the two projections for the sake of perspicuity inthe exposition : for it is easy to give to the construction allthe conciseness of which it is susceptible.

“ Second Method.— Figure 17. Let A, a, be the two pro-jections of the centre of the sphere ; ab, oraS, its radius ;BCD, the projection of its great horizontal circle ; and e f,ef, the projections of the given right line. Conceive theplane of the great horizontal circle produced till it cut thegiven right line in a certain point, and we shall have thevertical projection of the plane, by drawing the indefinitehorizontal line bag through the point a ; the point g, wherethis horizontal line cuts ef, will be the vertical projection ofthe point of coincidence of the plane with the given rightline ; and we shall have the horizontal projection, g, of thispoint by projecting g upon E F.

“ This done, take the same point for an apex, and conceivea conic surface covering the sphere, all whose generatinglines touch it in their respective points ; now, we shall havethe projections of the two horizontal generators of such conicsurface, by drawing from the point G, the two lines G c, G i>,tangent to the circle b c d, and which will touch it in the twopoints c, d, as may be easily determined. The conic surfacewill touch that of the sphere, whose line C d will be thediameter, whose plane will be perpendicular to the axis of the

ital P 1

i {O'

cone, and consequently vertical, and whose horizonjection will be the line CD. t0 tb e

If from the given right line two tangent p}j in 5. 9 ,, c h i ( ’

conic surface be conceived, each of them will fi t

according to one of the generating lines, which w> . lt1 ilthe same time on the conic surface and on the pi nn< ^f fli®since such generating line also touches the surface cird esphere in one of its points on the circumference of t |ie

projected in c D, it follows, that this point is at °” ce ,„'fii<i econic surface, on the plane which touches it, on t>'. e pt<rof the sphere, and on the circumference of the cii ^ a lljected in c D, and that it is a point of contact commo ^ t ,v0these objects. Hence we may conclude, 1st, I hat ^ (0 th 6planes tangent to the conic surface, are also tangen t(lC tsurface of the sphere ; 2dly, That their points o ( ,j,. c |ewith the sphere, being in the circumference of 1 ie p.gtprojected in c d, must be themselves projected on st * ] . 0 ughof this right line ; 3dly, That the right line passing () f tl‘ ethe two points of contact, being comprised in the P‘‘ lsame circle, must also be projected indefinitelyiarg e

“ We next proceed to an operation, for the plane ^jlurcircle parallel to that of the vertical projection, s ^ r | Z oi> tilthat which we have just finished for the great 1 p (; tb e

circle. The horizontal projection of such plane w j^iiiright line bah, indefinitely parallel tout; the Pj^proje® 16 *!it meets the given right line will be horizontally P ^, ,111“to the intersection, H, of the two right lines EF;.^. n , r tl.«its vertical projection will be obtained by I )r0 J. s url‘ lC f’point h upon e f in h. Conceive a new con ' n( jwhose apex shall be in this point of coincidence, ® ^ hA 6

■ath'g

tl,e >fat 6

like the former, shall cover the sphere ; and we B ". n(r jin 1the vertical projections of the two extreme generof such surface, by drawing from the point h, t° j nb k 1,- the tangents hn,hl, which shall touch *- c gU rpoints, it 1, as we may determine. This second c0 tl ewill touch that of the sphere, in the circumf ereI l c w j 1 jeli 1 Acircle, of which K 1 will be the diameter, and o ivjj

plane, perpendicular to that of the vertical proj e ,pp e c"consequently, be projected indefinitely upon tli«

cumterence of this circle will likewise passpoints of contact of the sphere with the

tang

required; whence the vertical projections of those | )tof contact will be somewhere upon K 1 ; and t > j e ci eby which these two points are united, will also j

upon the same line ki. po' nt9 |iy

“ Thus the right line drawn through the tw ve rti‘ ;! ^| econtact, is projected horizontally upon c D ’ a ” z0 nt»l c ''71;upon ki; it meets the plane of the great . ^

in a point, whose vertical projection is at the 1,1 . ^ nr 1 /of ki with bag, and whose horizontal P r0 J ecobtained by projecting the point n upon CD. c ird e ’ a ofl“ This done, suppose the vertical plane of ^ „s "K t |jjected in c D, to turn upon its horizontal diaine ^ a

1 . , _i i*> 1 w 11

■e' 1 V

l\1°

throng"

ent P iP tstwo ,-

a swivel, so as to become horizontal, and ^jj^oiig'it, in its movement, the two points of contact, ^ wlii cM

„e«

IWU points UI

its circumference passes, and the right line j j n h 5

" ,rcle ’ th® tb®

two points are

united. Construct this ci

diameter, in g

position, by describing upon c D, as a “‘““‘Jjne a" 1and if the position assumed by the ^ t |,e

sir®

nil

lior |!

tli® 1

J®

i" 6 . JiH* 1

r UP'

10®

Cl> DQ;

lerence c Vd be eons tructed, it win c«.

u P°n this '1, VV0 P°ints, which will deter:Position. 61 e nce considered

the swiv<d c d tIle h’ne of the two contacts,nient :■ this line not orange its position by 11 . „oi:‘

hash corf’ P“«* through^

oecome horizontal. Besides, the pen 1 * ,U