r

DE S

2.52

DE S

the latter lines intersect the correspondent arcs I k, p q, willbe the horizontal projections of the points of contact of thefirst surface with the tangent planes drawn along the givenright line.

“ The vertical projections of the same points will beobtained by projecting the points K, q, to k, q, on therespective horizontal lines i t, p s.

“ The horizontal and vertical projections of the points ofcontact being determined, the traces of all the tangent planesmay be constructed by methods similar to those alreadydescribed.

“ This mode may be easily generalized, and applied to anysurfaces generated by curves of determinate forms, andmutable as to their situation in space.”

Of the Intersections of curved surfaces.

“ When the generations of two curved surfaces are posi-tively determined and understood ; when the course of allthe points of space through which they pass is no longerarbitrary for either ; when for each of these points, one ofits two projections being taken at pleasure, the other may bealways constructed ; if these two surfaces have any points inspace common to them both, the positions of all such commonpoints is absolutely determined ; it depends on the form ofthe two curved surfaces and their respective positions ; andis of such a nature as to be always capable of being deducedfrom the definition of the generations of the surfaces, ofwhich it is a necessary consequence.

“ The course of all the points common to two determinatecurved surfaces, forms in general a certain curved line inspace, which, in very particular cases, may be found upona certain plane, having but a single curvature ; which, ininstances infinitely moi*e peculiar, may become a right line,without any curve ; and lastly, in cases still more rare, mayresolve itself into a mere point : but which, in the general,is what is denominated a curve of double curvature, becauseit ordinarily partakes of the curvatures of two curved sur-faces, upon each of which it is found at the same time, andforms their common intersection.

“ There exists between the operations of analyses and themethods of descriptive geometry, a correspondence, of whichit is here necessary to give an idea.

“ In algebra, when a problem is put into equations, andwe have as many equations as unknown quantities, we canalways obtain the same number of equations, in each ofwhich there enters but one of the unknown quantities ;whence we gain a knowledge of the value of each of them.The operation by which this is performed, is called elimina-tion, and consists in expelling, by means of one of the equa-tions, one of the unknown numbers from ah the otherequations ; so that by thus successively expelling all thedifferent unknown numbers, a final equation is obtained,containing only one, whose value it ought to produce.

“ The object of elimination in algebra bears a close analogyto those operations in descriptive geometry, by which theintersections of curved surfaces are determined.

“For example: suppose that in considering a point inspace, and representing, by x, y, z, the distances from thispoint to three rectangular planes between them, a relationbe established between these three distances, and that it beexpressed by an equation, wherein the three quantities x, y, z,and fixed quantities enter. By virtue of this relation, theposition of the point would not be determined; for the quan-tities x, y, z, may change their value, and consequently thepoint may vary in its position, without destroying the relationexpressed by the equation ; and the curved surface whichpasses through all the positions that the point may thus

occupy, without abating the relation established ^ at jo9the three co-ordinates, is that to which the e< Jbelongs. d by ■**

“ Thus, suppose a sphere, whose radius is express t [j«have its centre in the point of intersection com 111011 ^^three rectangular planes ; and that in considering ^ 9 up'point on the surface of the sphere, perpendiculars ^ taposed to fall from such point upon the three planes>^ jh®be represented by the letters x,y,z ; it is evidentradius of the sphere, directed to the point under ®°, pipe^’tion, will be the diagonal of a rectangular P ara e ff jll b®whose three aretes will be x, y, z; that its square . a t\&equal to the sum of the four squares of the four^are ^ y.that we shall thus have the equation a? + y p aC e 0Now, should the point change its position on the s _ - u lafthe sphere, its distances x, y, z, as to the three re £en tr®planes, would vary also ; but its distances as to 0 f itswould remain unaltered, and the sura of the sq ua tb®three co-ordinates, which is always equal to the ^^pere^!radius, would still retain its first value : we s ^ l0 p ,j ,|3 p°' ahave the relation between the co-ordinates eq 11 ®”

expressed by the equation a? + y 1 + z 1 = A*. 0 f tb®tion, which answers for all the points of the sur . „if,sphere, and for them only, is that of the surface i ^ t |,oiiljcurved surfaces have, thus, each its equation ; alit be not always easy to express it in such simpjjtain i* \as the distances x, y, z, it is always possible tomore complex quantities, such as the inclination 3 . j 0 r 0planes, or the radii of curvatures : but it is sufficie ^ oflpresent purpose to have conveyed our meaningexample. diff e | 9

“ If now, having, in x, y, z, the equations of * ^ 0 f *.curved surfaces, and supposing that for the P recia t>g' , ^ etwo surfaces the distances be taken to the sameplanes, we eliminate one of the three quantities^ 1 e>5latter, for example, from the two equations , tfb ,

first, by the similarity of these two equations) ^ t i,e pdoes not belong indiscriminately to all the P^^j^ch " e , | C 1)surface, nor to all those of the second, with y or ' v '* 1 , eoccupied, but only to those of their intersectio t j, e y ^each of the two equations ought to serve, b el g econ diy> ^ f,upon the two surfaces at the same time. t j 0 nequation in x, y, which results from the eh <Ji sta fry

expresses the existing relation between the tb® fl o

for all the points of intersection, whatever niay^ t ],ere 1 t j,etance z, which has disappeared, and about w ilonger any question in the equation ; it 0 f tb e

equation of the projection of the intersectiosurfaces upon the plane perpendicular to z. . n „. , tJI“ Hence we discover that in algebra, the ovVJ1 i»nation among many equations with three un ,j c ij allis to determine, upon the three planes to w gl irf a ®referred, the projections of the intersectionswhich the equations appertain. . e a naty? i0 tb®

“ The similitude between the operations con fi, ln. l !,the methods of descriptive geometry, are n g j tu ati°

instance just given, but prevails in Afon in ®P“ C 1working generating lines of any descripti or s g .ever movements be given to points, curve o pen lt ‘ 0 ? pythey may always be governed by analytie a e ypre 33 ®“ j, fl r gthe new objects which they originate, are paiiA t j 9 tbvery results of such operations : on the o ^ u t w'h , py 1no operation of analysis in three dimension^) ^ tj^

expression of a movement operated in s P aC .^ ta nce _ .1

VI M tuvi VU 1 VUV - - . rjijjv-

To obtain the most advantageous acquain ^mathematics, the student should early ac ® a 0 piperceive the existing analogy between 1

in 3 ® 1 fttion