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a ebnic surface contained within, and touching it, whoseapex is a point chosen at pleasure, being given in space; ifthe conic surface move without ceasing to be containedwithin, and to touch the curved surface, but so as that itsapex be kept in a right line, the plane of the curve of con-tact of the two surfaces will always pass upon one right line(which will be determined by the contacts of the surface ofthe second degree with the tw6 tangent planes that passalong the line of the apices); and if the conic surface moveso as that its apex be always in the same plane, the planeof the curve of contact will always pass upon the samepoint.

“ Second Question .—From a given point, to draw a planetangent to the surfaces of two given spheres.

“ Solution. — Figure 21. —Let a, a, be the two projectionsof the centre of the first sphere; b, b, those of the second ;and C, c, those of the given point. After having drawn theindefinite lines a b , a b, which are projections of the line thatwould pass through the two centres, and having constructedthe projections g e f, g ef hik , hik, of the great circlesof the two spheres parallel to the planes of projection, con-ceive a conic surface containing the two spheres within it,and touching them both at the same time ; the apex of thissurface will be in the line which passes through the two cen-tres. Then draw to the two circles g e f, hik, the twocommon tangents e n, f k, meeting in the point d, of theright line a b ; this point will be the horizontal projection ofthe apex of the cone, whose vertical projection, also, will beobtained by projecting the point d to d, on the productionof a b. Lastly, draw the projections c d, c d of the right linepassing-through the apex of the cone and the given point.Now, if from this latter line two tangent planes be conceivedto the conic surface, they will each touch it in one of itsgenerating lines, and consequently will both be tangent, atthe same time, to the two spheres. The question is thereforereduced to drawing two planes tangent to the surface of oneof the spheres, from the right line that passes through theapex of the cone and the given point; which may be done asin the preceding question, and the two planes will also betangent to the second sphere.

“ It must herd be remarked, that the same two spheresmay be supposed to be contained within two conic surfaces.The first conic surface will envelope them both on the out-side, and will have its apex on the outside of one of thespheres, with respect to the other, and the tangent planes toit will touch the two spheres on the same sides. The secondconic surface will also envelope both spheres, but will haveits apex between the two centres. The horizontal projec-tion, d', of this apex will be found by drawing to the circlesE F G, HIK, the two interior tangents, which intersect eachother in a point of the line A b ; and its vertical projectionby projecting the points i>' to d!, upon a b. The two tangentplanes, drawn to this conic surface will also each touch thetwo spheres; but they will touch the first on one side, andthe second on the other. Thus, four different planes mayanswer this question: for two of them, the two spheres areon the same side of the plane ; and for the two others, theyare on different sides.

“ Third Question .—To draw a plane tangent at the sametime to three spheres of given dimensions and positions.

“ Solution .—Imagine the plane tangent at the same timeto the three spheres; suppose, also, a conic surface contain-ing the two first of those spheres, and touching them both ;and the tangent plane will touch such conic surface in thewhole length of one of its generating lines, and pass throughthe apex of the cone. If a second conic surface be supposed,containing the first and third spheres, the same tangent plane

ting

nera 1 ’ 1 ' 0

will also touch it along the length of one of its S e y,lines, and will consequently pass through its apex- ^ng

if a third conic surface be conceived, enclosing a 11 ® g ti^the second and third spheres, the tangent plane j jj>oll gbtouch it along one of its generating lines, and pass .jj [#its apex. Thus the apices of the three conic surface 9 . n6in the tangent plane ; but they will also be in t gon t a in 9which, passing through the centre of the sphere, j an e9the three axes : they will therefore be in two diffe re ,^|^ jin 9 'at the same time ; consequently, they will be in a. ri= r0 jeC'Hence it follows, that if the horizontal and vertica ^ st j 0 P>tions be constructed, as laid down in the preceding ^ndof which two will be sufficient, the projection of a 11 the®*'upon the tangent plane may be made to pass throng^ ^ ye nThe question is therefore reduced to the drawing l' r0 ® s0 ugb*right line, a tangent plane to that of the three sph er j tl>' 3for, which may be done by the preceding methods,plane will be tangent t$ the two others. ijnag 1 ' 11 !

“ It must be observed, that, as we may alway 3 ^^ a n<jfor any two spheres, two conic surfaces which enV ut 5 jd®touch them both, the first having its apex on the th6< Wone of the centres, with respect to the othei, an< * t j ia t in ^having its apex between the two, it is evident, ^hicpreceding question there will be six conic surfaces, t' v °three will be on the outside of the three spheres,^ gpl> efe jby two, and three will have their apices between ^ anThe apices of these six cones will be distribute > v

three, upon four right lines, along each of wm tl'j^

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planes tangent at the same time'bus einrht different Dianes are sufhcie

for

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drawn two

spheres. Thus eight different planes arequestion : two of which touch the three spheres lt ,-side relatively ; and the other six are so dispose ‘ 0 tb e 'two of the spheres on one side, and the third on _

These considerations lead to the following propo 91 j n po s“ Figure 22.—Three circles whatever being g lV ^ tlV o

tion and magnitude on a plane, if, considering the ^ ^ the;

time, exterior tangents be drawn to them, produc g, 1intersect each other, the three points of intersecso obtained, will be in a right line. , j,;ch

“ Here, if we imagine the three spheres, or ^ ie3 - c

circles are the great circles, and a plane that ^ ree eOall three exteriorly, such plane will also touch t ‘ e tvv0 by b esurfaces contained within the spheres, considere ^ gge tl‘ ,and pass through their three apices, d, e, f < b ! ee ce at t )yapices are likewise upon the plane of the c0 nsefi l,etherefore they are on two different planes, anain a right line. tflr o, ’’’V'

If to the same circles, considered two tln' ee ,igl) 1

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tangents be drawn, intersecting each other, ^ jnpoints of intersection, G, h, i, will be, two by ^ ^ ?,line with the three first; so that the six points > ^

will be the intersections of the four right line foil®

“ This question is only a particular case o j»

which applies to all the three dimensions. t, e ing f'lir W“ The size and position of any four spheres^ eX teftl> espace, if the six conic surfaces circumscri a pice 3 ^joid

them, considered two by two, be conceived, ; n te rS ® t j 0 ii->

six cones will be in the same plane, and at c0ll icof the four right lines ; and if the other 91 e d‘ e

circumscribed interiorly by them, viz-, tha ‘ con ce ' ybetween the centres of the two spheres, ^ t jjj-ee h)apices of the six latter cones will be, ta >■ ^

in the same plane with three of the former- to“ Fourth Question .—From an arbitrary Ptangent plane to a given cylindric surface- traC Mt

“ Solution.—Figure 23. Let e i f k a tra cecylindric surl'ace on the horizontal pla ne ’

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