DES

249

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11 ^eta til 7 -

Pfoje C ti llle plane of the great circle, parallel to the verticalC . 0 ' n cidp 0tl ( a P°' nt whose horizontal projection is in tlie*>«al p ? ce > o, of the two lines cd, bah, and whose ver-is found by projecting the point o upon k i)?• c ' r cle m ' ts mov ement upon the swivel c d, a quarter of•toe o i . P® r P en dicular to CD, whose radius is the verticalI’etidipp’, ^en, a line be drawn through the point O, per-tra a8f eri . a \ t0 c D > and if upon such perpendicular o t be?>e ]; ne e< *> from o to T, the point t will be one of those of]j t)e contact, when it becomes horizontal. Therefore,°i Zinc'll 6 ^ rawn through the points n and t, its two pointstw Q l erice > p > Q, with the circumference cpdq, will behri* 0 ," 1 * 8 contact considered in the vertical plane,“ To tio ' vn -

joints in the horizontal projections of the same twobe rg . their natural positions, imagine the circle CD pqj^' v el q rne< ^ to Its original station, by turning on the same(Cribe ’ this movement, the two points p, Q, willV? C n T* ar ter-circles in the vertical planes perpendicular? lc ttla r ’ a l08e horizontal projections will be the perpen-U 0 p R S, dropping upon c D. The horizontal pro-P°n tb e .. he two points of contact will then be respectivelyo'^t lik e '? es p 9 and k s ; and as we have seen that they88 tty 0 , lse he upon c d, they must, consequently, be upon. “ Th e P °' nt8 coincidence K and s.

be P r °jections, r, s, of the same two points

\ ^hich a ' ne ^’ hy projecting the points b and s upon k i ;j Stt 'Cni j. ana ounts to the same, by transferring upon thea ° ' es Rr, s s, beginning from the horizontal line

9) , r , _O "_o

of *' ^he h^- Ua ^ to p E > an< l * s > equal to Q s.hi ^htact b'? onta l and vertical projections of the two pointsqj 8ln g constructed, the traces of the two tangenth, 8 °latioti ^ eterm l ne( l according to the method of the

S Tllis

by solution may be rendered much more con-

h(i °f th lI *“ t * le Pl anes °1" projection pass through theil 8 ^ 0 one g 8 ^* lere 5 which would reduce the two projec-

of* 1 ® 1 'ctn a pi! tt , er considerations lead us to the discovery ofd^hic ge .* e properties of the circle, of the sphere,ct ions, and of curved surfaces of the secondby ^ hag ^

8 8 pbe re e8E see n, that the two conic surfaces boundedt|| 9 ®» and (i each touch it in the circumference of aMu tv »o p 0 j* j ^eir circumferences would both pass tliroughL 88 - Thig 8 contact °1 the sphere with the tangent*hat ^ P ro perty is not peculiar to the two conic sur-

tk. ♦ ‘‘*6 CnnaiHciro^ Hut onnltaa tr\ nil ctinlt no lintrfi

ve considered, but applies to all such as have

jp le S'ven right line, and are circumscribed byby t | > havi n „ •’ ^Mrefore, we suppose a prime conic surface,ts tb' 8 8 Pbere . ltS a P ex upon the given right line, is boundedl °b« !lt ’ ts ape’ Un< * ^ this surface be supposed to move so* 8 ) r op 0tlta iiiert run along tlie right line without ceasingits " e< ? within, and tangent to the sphere ; it will inV f cle : a ][ '° n f touch the sphere in the circumference'be | JWo p 0 j n Which circumferences will pass through the1| 'tc 1 . 0 tang en f ln which the sphere comes in contact with|b« t 8<! t eaeij Planes ; and the planes of these circles will• N t | 0 8 ° n tii e t° t l<ir u P on one right line, which is that of^ili 1® giv<j n 8 ' : ^li" now the plane be conceived as drawn

.! 'i,

Per Pfl ' ,s

V,r

i r ‘ght line, and from the centre of the sphere,

!\1 l,8a tl;Cut

°Ue

t^ci

i Cll l a ° u oh the axes of all the conic surfaces, willU, ’ 1 a "by to t] t0 !' >e Pi anes °f aii the circles of contact,8t %,.i cut le r ‘ght line that is their common intersec-1 these planes

>pr

Point.

lines that will pass

° Clt y, a sphere and a right line being given,

if a number of planes, taken at pleasure, be conceived aspassing along the right line, which shall cut the sphere, eachin the direction of a circle ; and if, for each of such circles,a right conic surface be conceived, of which it shall be thebase, and which shall be confined within the sphere, theapices of all such conic surfaces will be another identicalright line.

“ By merely considering what happens in drawing theplane from the given right line, and from the centre ofthe .sphere, we are led to the two following propositions,which are immediate corollaries of what has preceded.

“ Figures 18 and 19.—Given in a plane, a circle, whosecentre is a, and any right line whatever, as b c ; if, afterhaving drawn from a point, as D, of the right line, twotangents to the circle, and the line e f connecting the pointsof contact, the point d be supposed to move along the rightline b c, drawing with it the two tangents, without ceasingto touch the circle ; the two points will shift their position,as well as the line E f, by which they are united, but thelatter will always pass through the same point, N, uponthe perpendicular, a g, dropped from the centre of the circleupon the right line b c.

“ And, reciprocally, if, through a point, as n, taken in theplane of a circle, a number of lines, as e f, be drawn atpleasure, each cutting the circumference of the circle in twopoints ; and if through these two points two tangents to thecircle, ed, f d, be drawn, to meet in a point, as d, allthe other points of intersection, found in the same manner,will be upon the same right line, B c, perpendicular to an.

“ It is not because all the points of the circumference areequally distant from the centre, that the circle possesses theproperty just described ; but because it is a curve ofthe second order, and has all its conic sections of the samequality.

“ Figure 20. —For example, let a e b f be a conic sectionof any kind, and CD any given right line upon its plane:suppose the curve to turn upon one of its axes, a b, to gene-rate a surface of revolution, and imagine the two tangentplanes to this surface drawn from the right line C D; the twoplanes will each have their particular point of contact. Nowtake for an apex, any point whatever, as h, of the rightline CD,and conceive the conic surface contained within, andtangent to the revolving surface, and it will touch the latterin a curve that must necessarily pass through the same twopoints of contact as the tangent planes. This curve will beplain ; its plane, which will be perpendicular to that of thegiven conic section, will be projected upon the latter, accord-ing to the line E f ; and this line will pass through the pointsof contact of the tangents with the conic section, drawnfrom the point H. Now, suppose the apex, H, of the conicsurface to move along the right line C D, without ceasing tobe contained within, and tangent to the revolving surface;in any of its positions, its curve of contact will possess thesame properties of passing through the two points of contactwith the tangent planes, of being plain, and of having itsplane perpendicular to the conic section. The planes, there-fore, of all the curves of contact will touch the ends of theline that unites the two points of contact, which is itselfperpendicular to the plane of the conic section, and hencethe projections of all the planes will be right lines that mustall touch the ends of the projection, n, of the line, by whichthe two points of contact are joined.

“ This proposition is only a particular case of anothermore general one, that takes place in the three dimensions,and which we shall be content with announcing in thisplace.

“ Any curved surface whatever, of the second degree* and