CURVE 



nitude; and as KM"M, KNM" are right angles, the 

 triangles KMM", KNM" are given in species : Hence 

 KN, NM" will each have a given ratio to M"K, and 

 consequently to MM". We have therefore KN=mz, 

 NM"=w z, (where m and n express given numbers,) and 

 as AH=AP— KN, and HK=PM"—M"N, we have 

 AH=x — mz, and HK=y—nz: Therefore these va- 

 lues of AH and HK being substituted in the equation 

 expressing the nature of the curve RKS, the result will 

 be the equation of the surface of the cylinder. 



Example. Let the curve RKS, the base of the cy- 

 linder, be a circle, whose centre is at A, the origin of 

 the co-ordinates, and its radius =a. In this case 

 AH«+HK 2 =a 2 ; therefore the equation of the surface 

 of the cylinder is (x — mzy + (y — nz) z =:a\ or 

 x" -\-y t -{- 2 m x z -f- 2 n y z -f- ( iri 1 -f. n ,£ )z z =a*. 



If the line KM be perpendicular to the plane XAY, 

 then m and n are each =0, and the equation of the cy- 

 linder is x 2 -f xf=a z . In general, the equation of the 

 surface of a cylinder perpendicular to one of the co-or- 

 dinate planes, is the same as that of its base or section 

 with the plane. 



To find the equation of the surface of a right cone 

 whose vertex is at A, the origin of the co-ordinates, and 

 the axis AR coincides with AZ, the axis of the ordinate 

 Z. LetiAPzrx, PM"=y, and M"M=z, be the co-ordi- 

 nates of any point M on the surface. Draw MR per- 

 pendicular to the axis of the cone, and join M"A. Be- 

 cause the cone is given, the ratio of RA to RM, that is, 

 of MM" to M"A is given ; therefore M"A=w z, (n being 

 a given quantity,) and since AM" 2 =:AP 2 -{-PM" 2 , we 

 liave for the equation of the cone 



It appears that the surfaces of a sphere, a cylinder, 

 and cone, have equations of the second degree, invol- 

 ving three indeterminate quantities. They therefore, as 

 well as innumerable other surfaces, may be all included 

 in one equation of this form 



A z 2 + B# 2 -f Cx*+Dy z+E x z+F x y 

 +Gr+H^+Kx+L=0. 



As lines are arranged in classes according to the or- 

 der of their equations, a like mode of classification' may 

 be applied to surfaces. Accordingly, on this principle, 

 a plane is called a surface of the first order, and all sur- 

 faces expressed by the above equation are said to be of 

 the second order. 



The equation of a curve surface admits of transforma- 

 tions perfectly analogous to those we have explained in 

 treating of curve lines : and they are effected in the 

 same manner, viz. by changing the origin and the di- 

 rection of the co-oi-dinates. Thus the general equation 

 to a surface of the second order may be changed to 



LINES. 



529 



x l -j- M y + N 2* 4. p = 0, 



where x, y, z still denote co-ordinates to perpendicular 

 planes. 



As a plane curve is produced by the common sec- 

 tion of a plane, and a curve surface of any kind, a curve 

 of double curvature may be produced by the section of 

 two curve surfaces. If a sphere and right cylinder, for 

 example, pass through each other, so that the centre of 

 the sphere is in the circumference of the cylinder, the 

 common section of their surfaces will be a curve of dou- 

 ble curvature. 



Let the centre of the sphere be at the origin of the 

 co-ordinates, and its radius =a, also let the axis of the 



VOL. VII. PART II. 



cylinder be in the plane XAZ, and put b for the radius 

 of its base ; then the equation of the two surfaces will 

 be 



x z +y* -f z 2 =a* ; 2 b x— x 2 =r# ! . 



And these two equations express the nature of the 

 curve under consideration. 



By eliminating y 2 from the first equation, we have 

 the nature of the curve also expressed by the equations 

 2bx-\-z l =a\ 2b x — x*=yK 



Of these two equations, the second is the equation of 

 the projection of the curve on the plane XAY, and the 

 ■first is the equation of its projection on the plane XAZ ; 

 so that instead of considering the curve as the common 

 section of a sphere and cylinder, we may regard it as 

 the common section of two right cylinders, one having 

 a circle for its base, on the plane of the co-ordinates x, 

 y, viz. that which we have supposed in the hypothesis of 

 the pioblem ; and another having its base on the plane 

 of the co-ordinates x, z. The equation of the surface 

 of this last being lb x-{-z z =za z , it is easy to see that 

 its base is a parabola. 



Whatever has been said in a former article respecting 

 the straight line, which is the common section of two 

 planes, will apply equally to curves of double curvature, 

 considered as the sections of two curve surfaces. Thus> 

 let z=F (x, y), z=f(x, y), be the equations of the sur- 

 faces, where F(.r, y), and f(x, y) denote any functions 

 or expressions of calculation made up of x, y, and known 

 quantities ; the co-ordinates of the curve, winch is then- 

 common section, must satisfy both these equations at 

 once. Also, by elimination, we may deduce from these, 

 three other equations, z=:<t>(x), z=<p(y), y=-^(x). These 

 give respectively the projections of the curve of double 

 curvature upon the planes of x, z, of y, z, and of 

 x, y ; each equation may be considered as belonging 

 to a cylindric surface, the base of which is upon the 

 plane of the two ordinates which enter into the equation. 

 It follows from thence, that there are always five curve 

 surfaces, any two of which may form, by then- intersec- 

 tions, one and the same curve of double curvature, so 

 that such a line may be formed in ten different ways. 



On the theory of curve lines, the following works 

 may be consulted, 



Des Cartes' Geomeiria. 



Newton and Stirling's Emimeratio linearum tcrlii or- 

 din is. 



Maclaurin's Geometria Organica. 



Maclaurin, De Linearum Geometricarum Traciaius, 

 end of his Algebra. 



De Gua's Usages de V Analyse de Descartes. 



Euler's Introduclio in Analysin Iufinitorum. 



Clairaut's Recherches sur les Courbcs a double Cour- 

 bure. 



L'Hospital Analyse des infinement pctils. 



Cramer Introduction u I' analyse des lignes courbes. 



Lagrange's Theorie des functions Analytiques. 



Waring's P/oprietates Atgebraicarum Curvarum. 



Du Sejour et Goudin's Traites des courbes algebriques. 



Biot's E:<sai de geomelrie analytique. 



L'Huillier Eiemens a" Analyse Geometrique. 



Monge's Application de /'Analyse a la Geomelrie. 



The Journal de I'Ecole Polytechnique, and the Cor- 

 respondence sur celte Ecole, contain many valuable me- 

 moirs on this subject. 



The general Properties of Curve Lines, by Emerson. 



(I) 

 3 x 



