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260

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“We shall now find each spherical surface cutting the firstsurface in the circumference of a circle whose plane is per-pendicular to the axis a a, and whose vertical projection maybe obtained by drawing the horizontal line h o, and its hori-zontal projection, by describing from the central point A, witha diameter equal to h o, the circumference of a circle kror'.In like manner,, every spherical surface of the series will cutthe second revolving surface in the circumference of a circlewhose plane will he perpendicular to the vertical plane ofprojection, and whose vertical projection may be obtained bydrawing through the point p a line, p n, perpendicular to a' b.

“ Should the points r, in which the two right lines ko,pn,intersect each other, be nearer to the two respective axesthan are the points kp, it is evident that the two circumfer-ences of circles would intersect each other in two points, ofwhich r would he the common vertical projection ; and acurve drawn through all the points r, constructed in a similarway, would be the vertical projection of the intersection ofthe two surfaces. By projecting the point r upon the circum-ference of the circle krob', in r, r', we shall have the hori-zontal projection of the two points of the coincidence of thecircumferences of circles found upon the same sphere ; andthe curve drawn through all the points r, r', constructed inlike manner, will be the horizontal projection of the intersec-tion required.

“ These examples may suffice for conveying an idea of themethod to be adopted in constructing the intersections of sur-faces, and drawing tangents to them ; more especially if thestudent be careful to make his constructions with scrupulousexactness, if he employ large dimensions, and, as much aspossible, trace the curves in all their extent.

“ In the foregoing pages, we have supposed the curves ofdouble curvature as being each determined by two curvedsurfaces, of which it is the intersection ; and, indeed, such isthe point of view wherein they most commonly present them-selves in descriptive geometry ; and, under this considerationwe have shown that it is always possible to draw tangents tothem. But, although a curved surface may be defined bymeans of the form and movement of its generator; it maynevertheless happen, that a curve may he given, by the lawof motion, from a generating point; in which case, if thepractitioner do not choose to have recourse to analysis, he mayadopt the method of Roberval. This method, invented byhim before Descartes had applied geometry to algebra, isimplicitly comprised in the processes of differential calculi,and is therefore not noticed in the elements of the mathe-matics ; a summary exposition of it in this place, will besufficient ; those who are curious to see numerous applicationsof this method may consult the Memoirs of the Academy ofSciences (French ) anterior to the year 1699, wherein theworks of Roberval have been collected.

“ When, pursuant to the law of its motion, a generatingpoint is constantly impelled towards one particular point inspace, the line it describes by virtue of such law is a rightline ; but if, in the whole course of its movement, it be atthe same time impelled towards two points, it will in generaldescribe a curve, though in particular cases it may describe aright line. The tangent to such curve may be obtained bydrawing through the point of the curve two right lines,following the two different directions of the motion of thegenerating point; by measuring upon these directions, in anappropriate manner, parts proportional to the swiftness of tworespective motions of the point ; by completing the parallelo-gram, and drawing the diagonal, which will be the tangentrequired: for this diagonal will be in the direction of themovement of the describing point to the point of the curveunder consideration.

“ The following is an example.

“ Figure 36.—A thread, a m b,

draw a tangent to it, by Roberval’s method. AJJ ,

as the length of the thread is not altered, the radi ^ ^

every instant of its motion, is lengthened in the s® ^ t her e '

,or*

tion that the radius b m is shortened. The swi

t'tness,

e qa

fore, of the describing point in the direction A M, j n gits motion in the direction mq. Therefore, '. , y. rn 1 e j l 'jj n es M f

a 0l> a

the ellipsis, the tangent divides the angle bM J :

one of the vector radii, and by the prolongation ot ^

two equal parts ; that,the angles AMS, B M R are eq ^ , e g e cti n »

otl‘ er *

of 9 :

other ; and that the curve possesses the property 01 _

upon one of its foci the rays of light emanating .gnsw- .

“ The method of Roberval in the case of three 1 ]C tion 0

-- -Itotheconstru

re- Thus, if

be constantly 1 er se,“■ win tfiU

though in some particular cases it may be a r ‘S'ipj iein general describe a curve of double curvature. , . e ve r >to such curve may be obtained in any point w ‘ g (Jir®^drawing right lines from the given point, in thetions of the movement of the generating point-; by^ pr 0 ? 0 ^upon such lines, in an appropriate direction, P a . nS of 1 ! etional to the swiftness of the three respective { (0

point ; by completing the parallelopiped, and tal ige [ltdiagonal of the parallelopiped, which will be tuthe curve in the point taken. analog 0 ^,,

“We shall now apply this method to a ease ^, e r®that of the ellipsis. The Figure 37, to wine io j eC tio n ’ 0represents the object in perspective, and not in P 9 c0 , * e , |0“ Three fixed points, A, b, c, being given • t ; e s tothread, a mb, be fastened by its two exti® n< j e nt t jjepoints a, b ; let a second thread, amc, indepe j g0 to ^ 0size of that of the other, be attached by its ex re^points a, c ; let a generating point, holding oo jt ^ g

moved so as to keep them in a state of Tn n orderdescribe a curve of double curvature. I n . 0 bser fe u gli'tangent to this curve in the point M, we mU . s ,- rD j th rO .. 0 iythe length of the first thread, a m b, being un ‘ ^out its movement, the part a m is lengthene s

in proportion as the part m b is shortened, an „AMi 9 1>ness of the generating point in the directiothat of its movement in the direction mb. i te ’ r ed, t^yjl t0length of the second thread, amc, being una ^ c ; 0 efi pfness of the generating point in the direction ^^rethat of its motion in the direction a a n( j on ( | 0 tiaf’measuring upon the prolongation of AM, 8 cof'P^ (1)®lines m b, m c, the equal parts M P, M Q, }I 0 bta’ n . 0( j,the parallelopiped mpusvqrt, we 8 ia req uir ;n0 ipl 0

■.thetangen , pi-ii> ^

founded, on ^

MPUSVQRT,diagonal, M S, of this parallelopiped,The method of Roberval being

A by 1Wbeing fasten® ^ ^ p 0 int,

extremities to two fixed points, A, b ; if, by means - g0M, this thread be stretched out, and the point m<W ^ scr ili®still to keep the thread in a state of tension, it g-yao

the curve d c m, being an ellipsis, whose foci ore ^ topoints a, b. From the generation of this curve, it ’^tano®^

A Ab

.sl t0

on

... ...V U..W..U.. 'C ...v.u.v-, , , , j.jlV

m b, and on the prolongation of A m, the equal rig 1 ' 1m p, and by completing the parallelogram M P R Q > 1 t |, e ge ne 'm r, of this parallelogram will be the direction o g ai»orating point in m, and consequently the tangent to t |, a tpoint of the curve. Hence wei may clearly P® rc ® |V £ ( ’ rlT1 e<l 7

in

may be readily understood, and applied to the c° ns }‘“~geprtangents to curves of double curvature r ui,ns. irating point move in space, so as totowards three different points, the line it

me metuou oi xvouervai ucjh r •— . e i,,.--

of compound motion, we may readily con ? oge n as e% g n cmore complex than these, which we have c ' to , ewe may avail ourselves of the known m e ^ w q,os® 9resultance of forces impelled towards a P 01 ’directions are ascertained.

tb e !

a* 1