ODE, 
 
ff^r^y^, 
 
 SCIENCE ATSTD ART DEPARtFM^lJc)p| 
 OF THE COMMITTEE OF COUNCIL ON EDUCATION, 
 SOUTH KENSINGTON MUSEUM. 
 
 A CATALOGUE 
 
 OP A COLLECTION OF MODELS OF 
 
 RULED SURFACES, 
 
 CONSTRUCTED 
 
 By M. FABRE DE LAGRANGE ; 
 
 WITH AN APPENDIX, CONTAINING AN ACCOUNT OF 
 
 THE APPLICATION OF ANALYSIS TO THEIR 
 
 INVESTIGATION AND CLASSIFICATION, 
 
 By C. W. MERRIFIELD, F.R.S., 
 
 PRINCIPAL OP THE ROYAL SCHOOL OP NAVAL ARCHITECTURE AND MARINE 
 
 ENGINEERING, AND SUPERINTENDENT OP THE NAVAL MUSEUM 
 
 AT SOUTH KENSINGTON. 
 
 
 r^ 
 
 ^^V 
 
 LONDON: 
 
 PRINTED BY GEORGE E. EYRE AND WILLIAM SPOTTISWOODE, 
 
 PRINTERS TO THE QUEEN'S MOST EXCELLENT MAJESTY. 
 
 FOR HER MAJESTY'S STATIONERY OFFICE. 
 
 1872. 
 
 29992. 
 
3^ 
 
 The set of Models described in this Catalogue is 
 deposited in the Educational Collection at the 
 South Kensington Museum, 
 

 Catalogue of a Collection of Models of Ruled 
 Surfaces, constructed by M. Pabre de La~ 
 grange, with an Appendix, containing an 
 account of the application of Analysis to 
 their investigation and classification, 
 
 By €. W. MERRIFIELD, F.R.S., 
 
 Principal of the Royal School of Naval Architecture and 
 
 Marine Engineering ^ and Superintendent of the 
 
 Naval Museum at South Kensington, 
 
 INTRODUCTION. 
 
 This collection illustrates the principal types of the class 
 of surfaces which can be traced out in space by the motion 
 of a straight line. 
 
 These surfaces, on account of the facility with which they 
 can be constructed and represented, and of the ease with 
 which their intersections can be determined, are of more 
 consequence than any others in the geometry of the Industrial 
 Arts. It is only in small work, which can be put into the 
 lathe, that the class of sm-faces of revolution approaches 
 them in respect of general utility. The most important 
 surfaces of all, the plane, the right cylinder, the right cone, 
 and the common screw, belong to both classes. 
 
 The representation of the surfaces by means of silk 
 threads is of course only approximate ; an approximation of 
 the same character as the representation of a curve by a 
 dotted or chain line, or by a series of right lines touching the 
 actual curve. 
 
 Fio. 1. 
 
Fig. 1 is an example of the first, and Fig. 2 of the second. 
 In both cases, the curve, although not actually drawn, is 
 indicated with sufficient approximation for most practical 
 purposes. Models Nos. 10 and 30 also afford illustrations 
 of the principle exhibited in Fig. 2. 
 
 The models are constructed with especial reference to the 
 possibility of changing their shape, by moving some of the 
 supports of the strings, by altering the lengths or positions 
 of certain parts, or by converting upright forms into oblique. 
 This possibility of deformation, as the process is technically 
 called, greatly enhances the value of the models, by allowing 
 them to represent a much greater variety of surfaces than if 
 they were fixed. They are, however, too delicate to be much 
 pulled about, and, unless they are very cautiously handled, the 
 strings are apt to become entangled or break. They should 
 never be used except by a person who understands them, and 
 they should not be shifted without some good reason. 
 
 In order to make this collection as useful as possible to the 
 student of geometry, it has been thought advisable to give, 
 in an appendix, a short account of the application of analysis 
 to the investigation of these surfaces, and of their properties. 
 The statement of these properties is scattered over a great 
 number of treatises and tracts, and there exists no single 
 work which gives a full account of ruled surfaces. The 
 appendix, of course, requires some knowledge of analytical 
 geometry of three dimensions. Any of the smaller modern 
 treatises, such as Aldis or Leroy, contain more than is 
 necessary as an introduction to the subject. The statements 
 in the appendix have been chiefly taken from Monge's 
 Applications de I ''Analyse a la Geometrie. 
 
 Geometrical drawings of most of the surfaces represented 
 by these models are contained in Bradley's Practical 
 Geometry (2 vols., oblong folio, published by Chapman and 
 Hall). Many of them will also be found in the French 
 treatises on practical and descriptive geometry, such as 
 Leroy, Adhemar, Lefebure de Fourcy, De la 
 GouRNERiE, and in their treatises on Stereotomy and Stone- 
 cutting {coupe des pier res). Many of them are also given in 
 Sonnet's Dictionnaire des Mathematiques Appliquees. 
 
CATALOGUE. 
 
 1. Hyperbolic Paraboloid generated by a single system 
 
 of right lines. 
 
 Two bars each pierced with holes equally spaced. One 
 bar is fixed, the other swings round an axis, which, moreover, 
 can be inclined at different angles to the fixed bar. 
 
 When the bars are parallel the strings indicate a plane. 
 When they are inchned to one another, but still in the same 
 plane, the strings still indicate a plane ; but when the bars 
 are not in the same plane, the surface is the hyperbolic para- 
 boloid. 
 
 This surface is sometimes called the twisted plane. But it 
 must not be supposed that it can be made by bending a plane. 
 On the contrary, when the surface is twisted, no two of the 
 strings lie in the same plane, and, therefore, no part of the 
 surface is plane. It can neither be flattened nor made from 
 a plane, without stretching or contraction. 'w.' 
 
 The hyj^erbolic paraboloid is the natural surface proper 
 for a ploughshare. 
 
 2. Hyperbolic Paraboloid. 
 
 Two bars pierced with holes at equal distances, the holes 
 being connected by two different systems of strings. The 
 surface, as well as the arrangement, is very nearly the same 
 as in No. 1, only that there are two paraboloids instead of 
 one. As the movable bar swings round, the paraboloid 
 opens out while the other closes up. If the bars are swung 
 so as to be in the same plane, one system of strings describes 
 a plane by parallel lines, and the other by lines radiating 
 from a point. If one bar is now turned so as to be end for 
 end, we still get a plane, the set of parallel lines now passing 
 through a point, while the set which previously passed 
 through a point has now become parallel. 
 
 The pair of paraboloids intersect in three right lines. 
 There is also a fourth intersection on the " line at infinity." 
 
 3. Hyperbolic Paraboloid. 
 
 Two bars, equally spaced ; each turns on an arm perpen*- 
 dicular to itself, and one arm swings on a pillar. These arms 
 can be ranged in one plane, and also turned end for end. 
 
4. llYFEa^BOLie. Pi^RABOLOiD generated by two systems of 
 
 right liiiGS. 
 
 A skew quadrilateral with four equal sides, each pierced 
 with the same number of holes, equally spaced. The model 
 exhibits the double generation of the surface. The plane 
 containing two of the sides turns about hinges connecting it 
 with the plane of the other two sides. By closing or opening 
 this hinge the paraboloid opens out or closes. When com- 
 pletely open, it forms a plane divided into diamonds. When 
 completely closed it again forms a plane, but the division is 
 no longer uniform. The strings then become tangents to a 
 plane parabola. 
 
 5. Hyperbolic Paraboloid. 
 
 A skew quadrilateral turning upon four hinges with parallel 
 axes or pins. 
 
 The difference between this and tlie last Is not in the kind 
 of surface or mode of generation, but in the manner of 
 deforming the surface. In No. 4 the lengths of the strings 
 alter ; while in this model they remain unaltered. More- 
 over, although the surface flattens in two ways, yet in both 
 ways the strings become tangents to a plane parabola instead 
 of parallel. 
 
 This model is well adapted for showing the leading sections 
 of the solid. All sections parallel to the pins of the hinges 
 are plane parabolas, which degenerate into right lines when 
 taken also parallel to the brass bars. Any other sections, 
 whether perpendicular to the hinges or inclined to them, give 
 hyperbolas, which degenerate into a pair of right lines when 
 the plane of section is a tangent to the surface. 
 
 It may be worth while to remark that there is nothing 
 absurd in the tangent plane to a surface cutting that surface, 
 as a student unaccustomed to those subjects might at first 
 think. On the contrary, when a sm-face is bent one way in 
 one direction and the other way in the opposite direction, 
 the tangent plane must cut it. In this case, the plane 
 passing through any two intersecting strings is a tangent 
 plane, and evidently cuts the surface along each string. 
 
 If we imagine two planes parallel to the hinge pins, 
 and each bisecting a pair of opposite bars, we obtain the 
 asymptotic planes of the paraboloid, each of w^hich is the 
 assemblage of the asymptotic lines of the hyperbolas parallel 
 to the principal hyperbolic section. Their being asymptotic 
 . has reference to these hyperbolas, and not to the parabolic 
 character of the surface. 
 
6. Hyperbolic Paraboloid. 
 
 A skew quadrilateral, with its opposite sides equal in 
 length, and pierced with holes at equal distances. 
 
 Nearly similar to No. 5, but differently mounted, and with 
 the sides of different lengths, the alternate sides only being 
 equal. It is virtually a slightly different aspect of the same 
 sufface as No. 5. 
 
 7. Hyperbolic Paraboloid. 
 
 A skew quadrilateral, with all its sides equal, and pierced 
 holes at equal distances. 
 
 As far as the curved surface is concerned, the same as 
 No. 5. But the hinges are altered in direction, and the 
 model shows plans and elevations of the right line generators 
 of the surface. The rings also show parabolic sections of the 
 surface. 
 
 In consequence of the alteration in the direction of the 
 hinges, the spacing of the inclined bars, although equidistant, 
 is at a different pitch from that of the horizontal bars. 
 
 8. Hyperbolic Paraboloid. 
 
 A skew quadrilateral with all its sides equal, and pierced 
 with holes at equal distances. It shows the plans and eleva- 
 tions of the right line generators. The rings show the 
 parabolas of the principal sections. 
 
 No. 7 represents one quarter of what is shown in No. 8. 
 The upper corners of Nos. 7 and 8 correspond ; but the lower 
 comer of No. 7 corresponds with the middle ring of No. 8. 
 
 9. Hyperbolic Paraboloid. 
 
 A skew quadrilateral with all its sides Unequal. The 
 surface is the same as Nos. 7 and 8, but the proportions 
 and the portion of the surface chosen for representation are 
 different. The quadrilateral base being Irregular, the strings 
 alter in length as the surface is deformed by closing the 
 hinges. 
 
 O" 
 
 10. Hyperbolic Paraboloid. 
 
 Skew quadrilateral, pivoting on a single hinge. Intended 
 to show the construction of the parabola connecting two 
 roads which meet obliquely. This constiiiction is used by 
 engineers in lading out roads. 
 
11. Hypeeboloid of one Sheet. 
 
 Two rings or circles in parallel planes are pierced with 
 equally spaced holes. In a certain position the threads give, 
 1st, a cylinder, and 2ndly, a cone. 
 
 The upper ring turns round a pin at its centre. In turn- 
 ing it, the cylinder closes in and the cone opens out, each 
 altering into a hyperboloid of one sheet. We can go on turn- 
 ing the ring until these coincide in one hyperboloid, of which 
 we thus get both systems of generating lines. 
 
 If the rings are set on a slope the hyperboloid is elliptic. 
 If the. rings are horizontal the hyperboloid is one of revolu- 
 tion. 
 
 Sloping one ring so as not to be parallel with the other, 
 gives rise to some curious iniled surfaces, but these are not 
 in general hyperboloids. 
 
 12. Hyperboloid of one Sheet. 
 
 Two rings of different radius in parallel planes are divided 
 into the same number of equal parts. The smaller and upper 
 ring turns round a pin at its centre. In a particular position 
 of the rings, the threads give two cones. Turning the ring 
 transforms each of the cones into a hyperboloid, and when 
 the two hyperboloids coincide, we get the two systems of 
 right line generators. 
 
 The same stand also has a model of a hyperboloid with 
 only one set of strings. By turning the upper ring either 
 way it deforms into a cone, in the one case with its vertex 
 between the rings, and in the other with its vertex at a con- 
 siderable height above the rings. 
 
 Both these can have their upper rings moved along the 
 top bar so as to incline the surfaces. We still get cones and 
 hyperboloids, but it is only when the rings are horizontal 
 and centre to centre, that we get surfaces of revolution. 
 
 13. Hyperboloid of one Sheet ; with its asymptotic 
 
 cone. 
 
 14. Hyperboloid of one Sheet ; with its asymptotic 
 
 cone. 
 
 The tangent plane to the cone is also drawn. It meets 
 the hyperboloid in two parallel right lincfit. 
 
 One of these right lines is the line of contact of a hyper- 
 bolic paraboloid witli the hyperboloid, and the tangent plane 
 is one of the director planes of the paraboloid, both systems 
 of generating lines of which are exhibited. 
 
9 
 
 15. Hypekboloid of one Sheet. 
 
 A slight variation from No. 14. The paraboloid only 
 shows one system of right line generators, and the tangent 
 ])lane is made by parallel instead of radiating lines. 
 
 16. Hyperboloid of one Sheet, and its tangent 
 
 paraboloid. 
 This shows the transformation of a cylinder and its tangent 
 plane into a hyperboloid and its tangent paraboloid. 
 
 17. Conoid with its director plane. The director curve is a 
 
 plane curve. 
 By shifting the position of the brasses, the conoids deform 
 into different conoids or other allied surfaces. 
 
 18. Conoid with a director cone. The director curve is of 
 
 double curvature. 
 By shifting the position of the brasses the conoids deform 
 into different conoids or other aUied surfaces. 
 
 19. Conoid showing both sheets of the surface. 
 
 By shifting the position of the brasses the conoids deform 
 into different conoids or other aUied surfaces. 
 
 20. Conoids. Model showing the transformation of a 
 
 cylinder into a conoid and back again. Also model 
 showing the transformation of a cone into a conoid and 
 back again. It is to be noticed that the head-lines of 
 the two conoids, that is to say, the right hne in which 
 the two sheets of each conoid meet, are perpendicular 
 to one another. 
 The transformation is effected by making the upper semi- 
 circle turn through two right angles. 
 
 21. Conoids. 
 
 Intersection of two equal conoids having a common director 
 plane. The horizontal intersection is a plane eUipse. 
 
 22. Conoid, in contact with a hyperboUc paraboloid. 
 
 23. Conoids. Two equal circles in parallel planes, divided 
 
 equi-distantly, are connected by threads, so as to form 
 four surfaces. 
 
 A cylinder. A conoid. 
 
 A cone. A second conoid. 
 
 The director planes, as well as the head Hnes, of these 
 conoids are at right angles to one another. 
 
10 
 
 24. Conoids. 
 
 Two equal circles in parallel planes are connected by 
 threads so as to form four surfaces. 
 
 A cylinder. 
 A cone. 
 A conoid. 
 
 A second conoid, with its director plane and line at right 
 angles to those of the former. 
 
 Same arrangement as No. 23, except that the lower ring 
 is replaced by a plane of section a little higher up. The 
 section gives, — 
 
 For the cone, a circle smaller than the upper ring. 
 
 For the cylinder, a circle of the same size as the upper 
 
 I'ing. 
 For the conoids, two ellipses turned crosswise. 
 
 25. Model exhibiting the simultaneous transformation of a 
 conoid into a cylinder, a cylinder into a conoid, the 
 paraboloid touching the conoid into the tangent plane 
 of a cylinder, and the tangent plane of a cylinder into 
 the tangent paraboloid of a conoid, and reciprocally. 
 
 The changes may be arranged as follows : — 
 
 From. 
 
 Conoid. 
 
 Tangent paraboloid. 
 Cylinder. 
 Tangent plane. 
 
 Into. 
 
 Cylinder. 
 
 Tangent plane. 
 
 Conoid. 
 
 Tangent paraboloid. 
 
 These changes are all effected simultaneously by one move- 
 ment, which can be reversed. 
 
 26. Model exhibiting the transformation, first, of a conoid 
 
 into a cylinder. Second, of the tangent paraboloid of 
 the conoid into the tangent plane of the cylinder. 
 
 27. French Skew Arch (biais passe). 
 
 The inner drum, of yellow thread, represents this surface. 
 It is a skew sm^face, with a right line du-ector ; and its faces, 
 the planes of the two semicircles, are usually parallel, although 
 the model permits them to be placed obliquely to one 
 another. The horizontal line joining the centres of the 
 two large semicircles is the right line director. 
 
 The construction for any one of the generating lines is as 
 follows : — Draw a plane tlu'ough the right line director at 
 
11 ' 
 
 any selected obliquity. It will, of course, give the radii of 
 the outside circles, and the line joining the points at which 
 it cuts the inside semicircles will be a generator of the 
 surface. This line will evidently pass through the du'ector 
 line, because it is in the same plane with it. 
 
 In stone or brickwork, the sides of the voussoirs will 
 be given by the auxiliary plane in question. When the 
 openings are parallel the voussoir joints are therefore plane, 
 and the simplicity thus gained is the chief reason for 
 adopting this form of skew arch. It is usual to take the 
 right line director perpendicular to the openings, and sym- 
 metrical to them ; that is to say, passing through the middle 
 point of the parallelogram of the springing plane. 
 
 When the openings are not parallel, the voussoir joints 
 shown by the model are deformed into hyperbolic parabo- 
 loids. This deformation, is, however, very slight, and in 
 practical work w^ould be avoided altogether by adhering to 
 the principle of drawing a plane through the director line. 
 
 The spacing of the voussou's is usually determined by 
 dividing the outer semicircle into equal parts. 
 
 This form of arch is inconvenient w hen the obliquity, and 
 the length of the barrel are excessive, for the generators are 
 not generating lines of the cylinder containing the opening 
 semicircles, but chords of it, and, therefore, at the middle, 
 falling considerably inside it. The arch therefore droops in 
 the middle, and this would be ugly and inconvenient if the 
 proportions were excessive. 
 
 It is interesting to compare this surface with the skew 
 vault of Marseille {arriere voussure de Marseille), an example 
 of wliich is shown in the set of plaster models contributed by 
 the '* Brothers of the Christian Schools," and another, in imi- 
 tation brickwork, among M. Schroder's models of fm-naces. 
 In this case the curvihnear directors do not tally with one 
 another, although they remain parallel, and the right line 
 director is a Vertical hue behind the smaller arch. The con- 
 struction for the right line generators is the same for both, 
 namely, to consider an auxihary plane pivoting about the right 
 line director. 
 
 28. Staircase Vault for a square well {vis St Gilles 
 carree), 
 
 29. Staieoase Vault. Model for exhibiting some pro-^ 
 perties of this ruled surface, by showdng how it is 
 obtained from the defonnation of a cylinder {douelle de 
 
 .la vis St, Gilles carree). 
 
12 
 
 30. Cylindek with Helix and DEVELorABLE He- 
 
 LIXOID. 
 
 The helix is simply a screw thread. The developable 
 helixoid, shown by the purple threads, is the surface swept 
 out by the right line tangents of the helix. If we consider 
 that each gore can be turned a very little bit about the thread 
 which separates it from the next gore, we see that the surface 
 can be flattened out or developed into a plane, without any 
 crumpHng. This happens because every two consecutive 
 generating lines meet one another on the heHx. That is why 
 its surface is called developable. Its section by a horizontal 
 plane is the involute of the circle. 
 
 The model allows the pitch of the heUx to be shortened 
 by lowering the upper plate, and the cyUnder can also be 
 incHned. When oblique, however, the curve which replaces 
 the helix is not such a screw thread as can be turned in the 
 lathe. 
 
 31. Skew Helixoid. 
 
 This surface is described by a right line which always 
 passes through the axis of a cylinder and makes a constant 
 angle with that axis. It also passes through a hehxor screw 
 thread traced on the cylinder. The model only shows the 
 surface, not the cyHnder. It is the surface of what is known 
 as the screw with a triangular thread. The section by a 
 horizontal plane is the spiral of Archimedes. 
 
 This is not the commonest form of the skew helixoid ; that 
 is best seen on the underside of a screw staircase, or on the 
 driving face of a common screw propeller. In these, two 
 generating lines are at right angles to the axis. 
 
 The surface may also be considered as generated by a line 
 which makes a constant angle with a given fixed line, and 
 moves up that line, and at the same time turns round it, at 
 uniform rates. 
 
 32. Skew Surface with its tangent paraboloid, capable of 
 
 transformation into another skew surface while the para- 
 boloid deforms into a plane. 
 
 This is (for a certain position of the lower semicircle) a 
 skew surface with a director plane, the plane being vertical* 
 The director curves are : one of them a circle divided equi- 
 distantly, the other a semicircle divided so as to keep the 
 strings parallel to the director plane. 
 
13 
 
 Intersections of ruled Surfaces. 
 
 33. Intersection of two cones having double contact with one 
 
 another, that is to say, having a pair of tangent planes 
 in common. 
 
 The consequence of their having double contact is that 
 their curve of intersection breaks up into two plane ellipses. 
 
 The vertices of the cones slide along a rule which turns on 
 a universal joint. See also model No. 38. 
 
 34. Common groin. Intersection of two cylinders having 
 
 a pair of common tangents. The model may be set 
 square or oblique. 
 
 35. Intersection of two cylinders, one piercing the other so 
 
 as to give two separate loops of intersection. 
 
 36. Intersection of two cylinders, having a common tangent, 
 
 80 as to give a curve having a double point at the point 
 of contact. 
 
 3Y. Intersection of two cylinders, neither completely piercing 
 the other, so as to give only one loop of intersection. 
 
 38. Intersection of two cones, having double contact, along a 
 
 pair of plane ellipses. 
 
 39. Groin. Oblique intersection of two splayed vaults of 
 
 the same spring. 
 
 40. A pair of intersecting planes, which, by pulling the brass 
 ball so as to give simultaneous rotation to the two upper 
 rods, deform into paraboloids first, and then into planes 
 described by radiating strings. 
 
 41. Intersecting cyhnder and plane. By pulling the brass 
 ball the head brasses rotate together, and the cylinder 
 deforms into, first, a hyperboloid, and then a cone, while 
 the plane deforms into, first, a paraboloid, and then again 
 into a plane with radiating hues. 
 
 42. A pair of intersecting cyhnders on circular bases. By 
 puUing the brass ball the head brasses rotate together, 
 and the cylinders deform, first, into hyperboloids, and 
 then into cones. 
 
 43. A pair of intersecting cylinders on irregular bases. By 
 pulling the brass ball the head brasses rotate together, 
 and the cyhnders deform, becoming at last cones. 
 
14 
 
 44. Groin. 
 
 Model showing the deformation of a common groin, both 
 obliquely, and by splaying the vaults. The model shows not 
 only the intersection, but the plans of the intersection and of 
 the generating lines. 
 
 45. Helix or Screw-thread. 
 
 Model showing the transformation of the right line gene- 
 rators of a right cylinder into screw threads of various pitch 
 or obliquity. 
 
 The pitch of p, screw is the distance between two succes- 
 sive turns, measured in a direction parallel to the axis. When 
 this distance is small, the screw is said to have a fine pitch, 
 when great, a coarse pitch or high pitch. 
 
15 
 
 APPENDIX. 
 
 By C. W. Merrifield, F.R.S., Principal of the Royal 
 School of Naval Architecture and Superintendent of 
 the Naval Museum at South Kensington. 
 
 This Appendix contains an account of the application of 
 analysis to the investigation and classification of mled eur- 
 faces. 
 
 It is not proposed to follow all the defoimations which 
 those surfaces can be made to undergo in the arrangements 
 illustrated by the models. That would take a very large 
 volume, and, even so, could hardly be given completely. The 
 analysis has been kept as simple as possible, and has been 
 written out in the form which appeared best adapted to the 
 consideration of surfaces, not with a view to their general 
 properties, but specially to the particular mode of generation 
 by means of straight hues. For that reason, no mention has 
 been made of the cones and cylinders of the second degree. 
 These are treated with sufficient fulness in all the ordinary 
 books. 
 
 The student may extend much of what is stated here by 
 introducing the principles of elUptic deformation and oblique 
 deformation. The latter is frequently equivalent to a change 
 in the direction of coordinates. Both these transformations are 
 applicable to nearly all that follows, and the student should 
 bear this application always in mind. He will do well to 
 work it out to its consequences in some of the simpler cases, 
 for which the formulae are not unmanageably long, as they 
 are apt to become if used injudiciously. 
 
 Motion of a right Line. 
 
 A right line is completely defined by the condition that it 
 shall meet four fixed curves in space, to the extent that there 
 is not an infinite number of right fines which will satisfy this 
 condition. The condition, in fact, gives rise to an equation, 
 which may have more roots than one, but in which each root 
 will have a definite and not a variable value. Moreover, the 
 root of the equation may be imaginary, and it may happen 
 
16 
 
 that the geometrical Hne may also be imaginary. Admitting 
 imaginary quantities, however, the line is definite. 
 
 If the hne be only conditioned to meet three curves in 
 space, it is free to move so as to trace out a curved surface, 
 v^^hich is called a ruled surface. It may, as before, happen 
 that the director curves are so chosen that no real hne can 
 meet all' three in real points. If, therefore, there are three 
 directors, one of them must be taken within certain Hmits of 
 position or direction in order that the problem may be really 
 possible instead of imaginary. The point is to see distinctly 
 that a right line must have three directors, and three only, to 
 trace out a surface. The surface so traced is called a ruled 
 surface. 
 
 It is possible to replace one of the director curves by some 
 equivalent condition. But the conditions taken altogether 
 must be such as are equivalent to the restriction imposed by 
 three director curves, neither more nor less. The classifica- 
 tion depends upon the selection of these conditions. Passing 
 by the common cone and cyhnder, let us proceed to consider 
 the next in order of simplicity. 
 
 The Hyperbolic Paraboloid. 
 
 This surface is traced out by a variable right hne, which 
 always meets three fixed right fines, which are parallel to one 
 plane.^ 
 
 Suppose that the plane in question is made to pass through 
 one of the lines, which is taken as the axis of x, and that this 
 plane is taken for the plane of (x, z). Suppose, also, that one 
 position of the variable line is taken for the axis of y. Then 
 the other two fixed lines may be written as — 
 
 y = *i \ y = ^h \ 
 
 and the surface must evidently contain these. But thes^. 
 equations give 
 
 1^X1/ -^ n^b^z — 
 l^ xy + n.;^h^z — 
 
 * This only introduces one condition among the director lines, for the plane 
 is arbitrary, and therefore may be taken parallel to any two given lines. It is, 
 therefore, only the third director which is restricted. 
 
17 
 
 and these will satisfy the equation cz — xy, if 
 
 "^ = "" IT ~ " IT 
 
 conditions which we may always satisfy by a suitable choice 
 of tiie axis of z. Moreover, it is clear that the equation 
 cz = xy will also be satisfied by the other system of lines 
 
 ar = « 
 
 my ■\- nz ■=■ 0. 
 
 . , , na 
 proviuecl c = . 
 
 ^ m 
 
 Hence the equation of the surface may always be written as 
 
 cz = xy, 
 
 whatever point of the surface be chosen as origin. 
 
 There is always one point on the surface for which, when 
 the equation is so written, z will be perpendicular to {xy) ; 
 but when that is so, it will not be generally true that x and y 
 are at rio;ht angles to one another. 
 
 <r5 to 
 
 If we leave the axis of z unchanged, and take, as new axes 
 of X and y, the bisectors of the old axes, we shall obtain a 
 rectangular equation of the form 
 
 £ _ ^-' _ t 
 an<l tlie old nxes will be given by 
 
 9 9 
 
 which is the equation to the pair of asymptotic planes of the 
 surface. When these asymptotic planes are at right angles 
 to one another, l=m, and the hy|:)erbolas, parallel to the 
 principal section, are rectangular. 
 
 In that case the equation cz — xy is referred to rectangular 
 axes, and we have the following curious properties, the 
 demonstration of which, whether by analysis or geometry, 
 will be a useful exercise for the student. Those marked (1) 
 and (4) hold, with slight modification, for the oblique case. 
 
 In the rectangular hyperbolic paraboloid cz=:xy^ 
 
 1. The areas of any two portions of surface which have 
 similar and equal projections on the plane of {xy) at 
 equal distances from the axis of z, are equal to one 
 another. 
 
 29992. B 
 
18 
 
 The inclination of the tangent plane is constant at any 
 
 given distance from the axis of z. 
 Any cyUndric annulus of the surface, with the axis of 
 
 z passing through its centre, may be measured by a 
 
 parabolic arc. 
 If the projection of a portion of surface be any figure 
 
 symmetrical to the projections of any two right Une 
 
 generators, the corresponding cyhndrical volume shall 
 
 be the product of the projected area into the ordinate 
 
 at its middle point. 
 Gauss's measure of curvature is constant at a constant 
 
 distance from the axis of z. 
 
 The Hyperboloid of One Sheet. 
 
 This in its most general form may be defined as the surface 
 traced out by a Une which meets any three fixed lines in 
 space. It will be simplest to take the three lines as three 
 edges (which do not meet) of a parallelepiped, to take axes 
 parallel to tliem, and to take the centre of the parallelepiped 
 Yis origin. Calling the lengths of these edges 2«, 2Z», 2c, their 
 equations may be written as 
 
 y = b, z= — c 
 z = c, X = — a 
 X = a, 1/ = — b 
 
 In order that an arbitrary line 
 
 may meet these three lines, we must have 
 
 b — ^ C + y 
 
 
 m 
 
 
 
 n 
 
 c 
 
 — 7 
 n 
 
 = 
 
 — 
 
 a -\- a 
 
 r 
 
 a 
 
 — a 
 
 = 
 
 — 
 
 h + ^ 
 
 
 I 
 
 n 
 
 multiplying these together we get 
 
 (« _ «) (/, _ ^) (^ _ ^) + (« + a) {h+ ^) (c 4- 7) = 0, 
 
 or, 
 
 a^y 4- bya + ca/3 4- abc = 0. 
 
19 
 
 And since (a, /3, y) is an arbitrary point on the line, the equa- 
 tion of the surface may be obtained by writing x, ?/, z, instead 
 of a, ^y 7, in this equation, that is 
 
 OT/z + izx + CXI/ + abc = 0. 
 
 The asymptotic cone is 
 
 ai/z + hzx -\- cxy = 0, 
 
 which may also be written as 
 
 a b c ^ 
 
 ^ + - + - = 0. 
 
 X 7/ z 
 
 These are the forms of equation which arise most directly 
 from the rectilinear generation of the surface. Another form 
 of the equation is 
 
 x^ y^ z^ ^ 
 a2 "^ P "" ^2 - 1. 
 
 the asymptotic cone then becoming 
 
 9 9 9 
 
 Let us try to follow analytically the deformation indicated 
 in model No. 16, starting, for simphcity, from the right 
 cyhnder. 
 
 Take for two of the axes the centre line of the cyhnder 
 and the line from its middle point to the middle point of its 
 line of contact with the tangent plane. Instead of supposing 
 that one ring only turns, it will be easier to suppose that we 
 turn them both equally, one backwards and the other for- 
 wards, through an angle = 6, Suppose, also, that for the tan- 
 gent plane we take a string whose distance from the point of 
 contact is tan ^ . 
 
 Before deformation the equation of the cylinder will be 
 .1-2 _|. y2 _ y,2^ ^^^ j^g tangent plane y = r. 
 
 The ends of the line of contact will be, at the top 
 a: = r cos ^, y = »' sin ^, 2" = c, 
 and at the bottom 
 
 X =z r cos ^, y = — 7' sin ^, 2: = — c. 
 The equations of the line of contact will therefore be 
 
 y _^ 
 
 r cos e r= 0, _ 
 
 rsm G c 
 
 B 2 
 
20 
 
 the square of the distance from the axis of any point at a 
 height z is therefore 
 
 .T^ + 2/- = 7'^ cos^ B -h -^ r^ sin^ 0, 
 
 and this is the equation of the surface. 
 
 Now with reference to the string in the the tangent phvne ; 
 before deformation, its distance from the axis will be r sec f ; 
 after deformation^ the position of its upper end will be 
 
 X = r sec (p cos (^ + 0), y = r sec <p sin (^ + B), z = c, 
 and of its lower end 
 
 X = r sec ^ cos {<p — Q), y = r sec ^ sin (^ — e), r = — r. 
 Hence its equations will be 
 
 X — r sec (p cos (f + 6) 
 
 z — c 
 2c 
 or, 
 
 1' sec 
 
 <p {cos (f + ^) — cos (? — e)} 
 
 y cos 
 
 <p — T sec (p sin (^ + 0) 
 
 r sec ^ 
 
 {sin (0 + ^) — sin {<p —0)} ~ 
 
 X cos 
 
 ^ — r cos (p + e) 
 
 
 r sin <^ sin 
 
 _ // cos ip 
 
 - r sm {<p -\- e) __ z — c 
 
 r cos ^ sin ^ c ' 
 
 We easily find from these 
 
 II z . 
 
 sin = tan ^ cos 
 
 re ^ 
 
 X z ~~ 2c 
 
 cos 6 = tan q fin 6 
 
 r c ^ 
 
 and we can eliminate tan p by simple division. This gives 
 us the equation of the surface, since ^ is a parameter which 
 indicates only the position of the line in the surface, a hyper- 
 bolic paraboloid. 
 
 It will be an easy and useful exercise for the student to 
 find the other system of generating lines of the paraboloid, 
 and to show that they are horizontal. 
 
 The leading properties connected with rectilinear genera- 
 tion are as follows : — 
 
 A tangent plane to the liyperboloid cuts the surface in two 
 right lines, which intersect at the point of contact. 
 
21 
 
 A tangent plane to the asymptotic cone cuts the hyper- 
 boloid in a pair of parallel Knes. 
 
 A plane parallel to a tangent plane of the asymptotic cone 
 cuts the hyperboloid in. a parabola. 
 
 Conoids. 
 
 The simplest form of this surface is when a line moves 
 parallel to the plane of (x, y) and always passes through the 
 axis of z, passing also through a circle witli its centre on the 
 axis of X, and its plane at right angles to that axis. This is 
 what WalHs named the cono-cuneus. Call the radius of the 
 circle c, and let a be its distance from the origin. Then for 
 any particular value of z, say z = h, the equations of the 
 
 ?/ 's/ c^ —. li^ 
 
 generatmg line will be 2; = k, and -. = 
 
 X c 
 
 The equation of the surface therefore will be 
 
 y = ^ ^^ - ^'\ or 
 
 X c 
 
 C2 (^2 _ ^2^ + ^2 ^2 ^ 0, 
 
 The sections at risjht ano-les to the axis of x are therefore 
 elliptic. Those parallel to the plane of x, y are evidently 
 pairs of right lines. Other plane sections of the surface are 
 in general curves of the fourth degree. 
 
 This particular case sufficiently exhibits the characteristic 
 form of the conoids. There is more complexity, but no real 
 difficulty in obtaining the equations of the conoids described 
 under other conditions. They are chiefly met with in the 
 case of splay arches. 
 
 Models Nos. 25, 39, and 44 are examples of this applica- 
 tion of the sm-face. 
 
 Families of ruled Surfaces. 
 
 The general consideration of any class of ruled surfaces in 
 which one or more of the dii*ector curves are left arbitrary, 
 requires the use of the arbitrary functional symbol. This 
 symbol can only be got rid of by partial diffisrentiation. 
 
 Cylindrical Surfaces. 
 
 A right line moves always parallel to itself. Its equations 
 will therefore be 
 
 a.* — a__y — ^__2r-~7 
 I m 
 
22 
 
 where l, 711, n, arc fixed quantities known, if the direction 
 of the generating line be known ; but a, /3, and 7 are variable 
 quantities depending upon the director curve. We may write 
 the equations as 
 
 X y a ^ 
 
 I m~~ I m 
 
 m n m ii 
 
 , , u B y 
 Of the three quantities ~j ~ ~i one is absohitely arbitrary, as 
 
 we only have to do with their differences. We may there- 
 fore consider 
 
 to be a function of 7 
 
 m n I m 
 
 the particulnr form of the function depending u])on the 
 character of the director. We may therefore write the 
 equation of cylindrical sui-faces as 
 
 L 111 \in n J 
 
 in which F is a functional symbol. 
 
 Tt might seem at first sight that tlie common equation of 
 a right cylinder 
 
 is not of this form. But any chnnge of variables such as 
 will introduce the third variable z will nt once reduce it to 
 this form. 
 
 Conical Surfaces. 
 
 A right line always passes through a fixed point. Calling 
 the point a ^ 7, its equations are 
 
 a: — a_y — /3_-2; — 7 
 / m n 
 
 whence 
 
 X — a, _ I y — ^ ^m 
 
 y — i^~~m'z^y n 
 
 I 711 
 
 and — must be some function of — . This gives 
 
 m 
 
 y - ^ \z- yJ 
 
23 
 
 as the equation of conical sui-faces. Its form expresses 
 simply that x — a,y — ^^z — y are connected by a homo- 
 geneous equation. 
 
 Conoids, 
 
 A right line is conditioned to pass through a fixed right 
 line and remain parallel to a fixed plane. Take the fixed 
 line for the axis of r, and the fixed plane for that of (x,i/). 
 Then the variable line may be written as 
 
 X -\- ki/ = z ■=■ c, 
 
 where c and k are variable, and are evidently connected 
 together by some equation depending upon a second director. 
 We shall, therefore, have k = F(c) = F(e), and the equation 
 of the surface is 
 
 x + y Y{z) = 0. 
 It may also be written as 2: = / ( — ) . 
 
 If the director line and plane be taken arbitrarily, a change 
 of coordinates will be necessary. But the effect of this will 
 only be to put the equation in the form 
 
 f a,x + b.^y + c^z + 4 \ 
 a.^^Wy + c,z^.e,=j [a,x + h,y -, c.,z + d)' 
 
 The hyperbolic paraboloid is a particular case of the 
 conoid. So also is the common form of the skew helixoid, 
 or screw surface, not the one illustrated in this collection. 
 
 Ruled Surface with Director Plane. 
 
 This is more general than the conoid, inasmuch as it is 
 not restricted to pass through a right fine director. The 
 removal of this restriction introduces another arbitrary 
 function. Taking the dh*ector plane through the origin and 
 caUing it 
 
 ax -{• by + cz =1 0, 
 
 the equations of any line parallel to it may be written as 
 
 ax -{- by + cz = k, Ix + my = 1, 
 
 where k, I, and m are arbitrary constants. We may regard 
 I and m as functions of k, and we get 
 
 xF {ax + by -\- cz) -f yf{ax -\- by •{■ cz) = 1, 
 
24 
 
 Monge writes it in the form 
 
 z = xF (ax ■\- hy -\- cz) + yf{tcx -\- by -^r cz) 
 
 which comes to the same thing, as F and / are arbitrary 
 functions. Monge's form may be obtained dh*ectly by 
 taking the equations of the line as 
 
 ax -{- by -\- cz = k, Ix -\- my = z. 
 
 if the plane of xyhQ taken for the director plane, a and b 
 A'anish, and we may write the equation as 
 
 xYz + yfz = 1 or z. 
 
 Ruled Surface with right line Director, 
 
 This again requires two arbitrary functions. Taking the 
 line for the axis of z, the projection of the variable line on the 
 plane of {x, y) \vill be of the form 
 
 X — ky •=. 0, 
 
 ^vhile its projection on the other plane, say of (x, z), will be 
 of the form 
 
 mx -\- n =■ z. 
 
 Now making m and n arbiti'ary functions of k ov — we have 
 for the equation of the surface 
 
 -"•(1) + /(!)■ 
 
 To this family of surfaces belong the hyperboloid of one 
 sheet, all conical surfaces, conoids, skew hehxoids, the skew 
 arch, known by the French name of the " biais passe ^^ 
 (Model No. 27 of this series), and the splay vaulting, known 
 as the " Arriere voussure de Marseilhr Of the last, there 
 are two specimens in the education museum at South Ken- 
 sington, although it is not represented in this series. 
 
 The ruled surface with a dnector plane is the particular 
 case of the mled surface with a right hne director, in which 
 the right hne has moved off to infinity. It follows that 
 conoids are a particular case of surfaces with two right line 
 directors. These du-ectors must not meet in a point, or the 
 surface will reduce to a plane. 
 
 Ruled Surfaces with two right line Directors, 
 
 The most symmetrical form in which the two du-ectors can 
 be referred to rectangular axes is to take the shortest distance 
 
26 
 
 between them for one axis?, and its middle point for origin, and 
 to let the other axes bisect the angle between the two dii*ectors, 
 whose equations are thus — 
 
 X = a, ?/ =:: z tan 
 X =. — a,y = •— 2; tan 0. 
 
 Thus the line joining a point (2 = k) on one to a point 
 (z = l) on tlie other is 
 
 X — a _ y ^ k tan d _ z — k 
 
 2a {k + 1)timB k — I 
 
 Whence A = -^, .^-+^* 
 
 tan B X i- a 
 
 a y — z tan B 
 tan X — a 
 
 in which k and I are parameters, one of which may be assumed 
 
 to be a function of the other. It follows that is a 
 
 a 
 
 function of — ^ . Hence the functional equation of tiie 
 
 sm-face is 
 
 ?/ -f z tan B __ TjT /i/ "" 2; tan B 
 X -f a \ X — a 
 
 )■ 
 
 This is not the general form of the functional equation, but 
 its simplest form, analogous to that which we obtained for a 
 conical surface, by assuming the position of the vertex to be 
 at the origin, or to that of the ruled sm-face with one right 
 line director when we take that for an axis. 
 
 Another form, even more symmetrical, can be obtained by 
 taking the origin and axis of x as before, and lines parallel to 
 the dnectors as (oblique) axes. The equations of the directors 
 are thus 
 
 X =^ a, z = 
 X = — a,y =0 
 
 and the Une joining a point y = k on the first with a point 
 2: = / on the second, is 
 
 x^a 'i 
 
 1 — k z 
 
 2a ~ 
 
 k ~ I' 
 
 / z 
 
 k y 
 
 Whence ,, — ,0 
 
 2a a — x^ 2a a -\- x 
 
26 
 
 and the functional equation of the sui-face is 
 
 = F 
 
 y V 
 
 a — X \a -\- xl 
 
 Here, however, the coordinates are oblique. 
 
 The ruled surface with two director planes is cylindrical. 
 
 Ruled Surfaces generally. 
 
 It is not possible to express the general equation of ruled 
 surfaces in a purely functional form, for the equations of 
 a right line are only two in number, and they involve four 
 arbitrary constants. We cannot clear all four by such sup- 
 positions as we have made in the previous cases without 
 introducing some similar restriction. If, for instance, we 
 take 
 
 a^ =. y -\- h 
 
 z =. ex -\- e 
 
 and consider c and e functions of a we get 
 
 in which h remains perfectly arbitrary, and as it is not a 
 function of x and y alone, we cannot get rid of h without 
 bringing back one of the other arbitrary quantities or intro- 
 ducing a restriction. We cannot expunge them without 
 partial diiferential equations. 
 
 It is also to be remarked that the functional equations 
 which we have hitherto obtained are not quite general. In 
 the case of cones, for instance, we have given the equation 
 
 , = F ( ^- ) on the assumption that the vertex is at 
 
 y — \z — cJ 
 
 the known point, whose coordinates are (a, b, c). If the 
 vertex is to be left arbitrary, so that the functional equation 
 shall express all conical surfaces without reference to the 
 position of the vertex, no such functional equation exists, 
 nor can the condition in general be expressed by any one 
 differential equation. 
 
 Developable Surfaces, 
 
 The characteristic of these surfaces is essentially differen- 
 tial and it cannot be expressed without partial differentials. 
 
27 
 
 Surfaces of ivhicli the Directors are given, either explicitly 
 or hnjilicitly. 
 
 In these cases the arbitrary functions must be determined 
 by the conditions in question. In the case of ruled surfaces 
 with given director lines, through which the variable right 
 line must pass, it will in general be sufficient, for the deter- 
 mination of the arbitrary function, to make the equations of 
 the director satisfy the functional equation identically, for 
 then the director wiU be a line traced upon the surface. 
 
 Differential Equations of Families of Surfaces. 
 
 In the restricted cases in which we have obtained functional 
 equations, we may obtain the differential equations by the 
 direct processes of partial differentiation. But they may also 
 be obtained from the equation to the generating line, expressed 
 in terms of the coefficients which we wish to retain ; for our 
 object is simply to obtain the relation between certain partial 
 differential coefficients, and it is immaterial whether we obtain 
 these by implicit differentiation from a line in which only one 
 variable is independent, or by partial differentiation from a 
 surface in which there are two independent variables. 
 
 The character of the restriction imposed on the motion of 
 the line will determine what are the constants to be eliminated 
 by the differentiation. For the motion of a point fixed while 
 the line shifts, and the independent motion of the point along 
 the line, each infinitesimal in amount, determine the tangent 
 plane of the surface, and this tnngent plane is one of the 
 complete solutions of the partial differential equation, of which 
 the functional equation of the surface is the general solution. 
 
 Any family of surfaces of which the actual surface is the 
 envelope will be a complete solution of the differential 
 equation of the surface, but the tangent plane is the only 
 one in which we are certain that the right line generator 
 must be contained. 
 
 Cylindrical Surfaces, 
 Starting from the right line 
 
 X — a. _ y — ^ _^z — y 
 I VI n 
 
 and observing that our object is to retain l^ m, n, we obtain 
 by implicit differentiation 
 
 dz dz dy n dy m 
 dx dy dx I dx I 
 
28 
 
 therefore 
 
 , dz dz 
 
 t - -j- m -J- = u 
 ax dy 
 
 which is the differential equation of all cylindrical tjurfaces 
 whose director ratios arc l, m, n. 
 If we take the functional equation 
 
 n in \rn I I 
 
 we obtain by partial differentiation 
 
 ^ = _ Z? F' ~ = ~ 4. ?i F' 
 dx I ' dy m m 
 
 .'. I J- + m -y- = ?^ as before. 
 ax dy 
 
 Conical Surfaces. 
 
 Our object here is to retain a, 1^, 7, and to get rid of 
 l^ m, n, by differentiation. For this purpose write 
 
 z — y _^n y — ^ _m 
 
 y — ^ m X — a,~ I 
 
 (■^- - -) I - (y - ^) = 0. 
 
 , ^. ( dz ^ y — ^ dz ) , . y 
 
 X — ex. 
 
 or 
 
 / ■ dz , .. dz 
 
 is the differential equation of the surface. 
 
 Co)w'ids, 
 
 The generating line is, for the simplest form, 
 
 y = kx, z = c. 
 
 . dz _^dz dy^ ^^ ^ = /, = ^ 
 ' ' dx dy dx ' dx x 
 
 eliminating --^, the partial differential equation of the sur- 
 face is dz ^ dz 
 dx dy 
 
29 
 
 Ruled Surface with director Plane. 
 
 Let the equations of the line be 
 
 Ax + By + C^ = /c ( ^ .) 
 
 ax + hi/ = I (2.) 
 
 in vv'hich a, b, and ky are r.rbitraiy coiivstants. Differentiating 
 implicitly 
 
 a + l% = 0. (4.) 
 
 Proceeding to a second differentiation, in which, by virtue 
 
 dy 
 of the last equation, we may consider ~ as constant, we 
 
 obtain 
 
 dx'' ^ dxdy \dx) ^ dif \dx) ' ^^ 
 
 Then, if we take (2) as the director plane, that U to say, if 
 we make the director plane parallel to the axis of z, we must 
 
 substitute in (5) the value of -/ obtained from (4), and we 
 ^ dx 
 
 get 
 
 j^dh c^ J dh , 9 d^z . 
 dx^ dxdy dy^ 
 
 or in the usual notation 
 
 Wr - 2ah8 + f/^Jf = 0. (6.) 
 
 If, however, we take ^ (I) as the director plnne, we must 
 
 determine j ^^^"^ (•^)j which gives 
 
 dif _ _ C/? + A 
 llx ~ Cq + B 
 
 and the differential equation becomes 
 
 (C^ -f B)2r - 2 (C7 + B) (C> -F A) 5 + (C;7 n Aj^ f 
 = 0. (7.) 
 
 which reduces to the same form as (()) if C r= 0. 
 
30 
 
 • Ruled Surface with right line Director, 
 
 The axis of z being taken as the director, the equations of 
 the variable line may be written as 
 
 whence 
 and 
 
 X =^ ky mx -\- n = z. 
 
 dy \ y 
 
 j^ = -,- = — = constant, 
 
 ax k X 
 
 _ dz dz dy 
 dx dy dx 
 
 Diiferentiating again, 
 
 = ^+2^^ + ^^^^ 
 dx'^ dxdy dx dy'^ \ dx) 
 
 t , y dij 
 Substituting - for -j- we obtain 
 '^ X dx 
 
 „ d^z ,^ d^z „ d^z ^ , ^ 
 
 which is the equation of the surface. 
 
 If we had taken any other right line than the axis of z for 
 the director, we should have had the constants of its equa- 
 tions appearing in the differential equation (as in equation 
 No. 7 of the previous section). Equation (8) is therefore 
 not a general one, but a restricted form, in which the con- 
 stants have received particular values which simplify the 
 result. 
 
 Ruled Surfaces considered generally. 
 
 We have seen that we cannot express the equation of a 
 ruled surface in a functional form, without differentiation. 
 This happens because the simplest expression of a ruled sur- 
 face is, that its tangent plane at any point shall meet it in a 
 right line, or, what is the same thing, that one of its tangent 
 lines at any point shall be wholly in the surface. Now the 
 question of tangency is emphatically a question of differen- 
 tiation. 
 
 In what follows it will be convenient to use the ordinary 
 abridged notation of partial differential coefficients, namely, 
 
 _^dz _dz 
 
 ^ ~~ dx ^ ~~ dy 
 
31 
 
 dx^ ' ~~ dxdy ~~ dy^ 
 
 ~~ dx^^ ~ dx^dy 
 
 d^z ^-^ 
 
 ~ dxdy'^ ~ dy^ 
 
 A ruled surface is expressed with complete generality by 
 considering it to be traced out by the motion of the gene- 
 rating line 
 
 'O 
 
 y = c^x + c^ 
 
 in which the four quantities c, which are constant so far as 
 the equation of the line in any one position is concerned, but 
 variable parameters, when considered with reference to the 
 position of the line, are to be made to disappear. The 
 obvious way of effecting this is to obtain, by means of implicit 
 differentiation, a relation between the partial differential co- 
 efficients of 2, which, when thus cleared of what is special 
 to the particular generating line, will be tlie differential equa- 
 tion, in partial differentials, of the surface. 
 
 Operating implicitly upon the above equations we get 
 dy dy 
 
 whence 
 
 p -\- qc^ = q (1.) 
 
 Differentiating again upon the same suppositions we have 
 
 r + 25^2 + tc,;^ = 0. (2.) 
 
 Differentiating a thkd time, we get 
 
 a -f 3/3^2 + 370.2 + ^c.^ = 0. (3). 
 
 If now, by ordinary algebra, we eliminate c^ from equations 
 (2) and (3), we get the ditferential equation of ruled surfaces. 
 It is not worth while to write it down, as it is more conve- 
 niently used in the impHcit form given above, than in its very 
 cumbrous expHcit form. 
 
 We liave already noticed that the general form of a ruled 
 surface cannot be expressed as a single functional equation. 
 It follows that the differential equation has no general 
 primitive. 
 
32 
 
 Developable Surfaces, 
 
 If we consider a stiff card, of the form a X Yf E in the 
 figure above, to be deeply scored along the right lines B A h, 
 C B c, D C f/5 E D e, so that we can bend it along each of 
 them, the broken line ABODE will form a polygon, at 
 ■first plane, but a skew polygon when we come to bend the 
 the surface, which will then form a polyhedron, every edge 
 of which will run into every successive edge, along the 
 broken Kne ABODE. It is evident that this condition is 
 necessary to our being able to deform the surface. For if 
 one of the scores (say c B) stopped short of the edge ABO, 
 the card would not bend, and if A B were not an actual 
 edge, in that case it would not bend either. 
 
 Now if we consider a surface which can be formed by the 
 gradual bending of a plane surface, the only departures from 
 this type are, — 
 
 (1.) That the polyhedral surface is replaced by a curved 
 
 surface. 
 (2.) That the polygon A B D E is replaced by a cui^e. 
 (3.) That instead of a finite bending along a few lines 
 A «, B J, c, &c. we have an infinitesimal bending 
 along an infinite number of such lines infinitely 
 close together. - 
 (4.) That all these lines are tangents to the curve ABO... 
 
 which replaces the polygon. 
 (5.) That the consecutive lines A «, B Z>, meet one another ; 
 that is to say, the shortest distance between them 
 is an infinitesimal of a higher order than the dis- 
 tance between any other two points of them. This 
 is imphed in their being tangents to the same curve. 
 
33 
 
 If the curve ABODE degenerates into a single point, 
 there will occur some singularities which will mask the 
 general properties of these surfaces. This degeneration gives 
 conical surfaces. When, further, the single point in question 
 moves off to infinity, we get the cylindrical surfaces. 
 
 This is not all. If we take two cards, counterparts of 
 one another, and glue them together along the polygon 
 A B C P E, we can deform the double plane into a double 
 polyhedron, the two sheets of which will be placed back 
 to back. There will then be a finite angle between the 
 corresponding gores in the two sheets. When we replace 
 this polygon by a curve, this angle will be infinitesimal, and 
 a section at right angles to A B C D E will be a cusped 
 curve. 
 
 Each sheet will evidently be a ruled surface, the variable 
 right lines of which will touch the curve which replaces the 
 polygon ABODE. It is evident that the right line 
 generators must otherwise stop abruptly, for they could not 
 get from one sheet to the other on any different conditions. 
 
 A good imitation of this can be made by taking two 
 sheets of paper, fitted over one another, as A B D, cutting 
 out a curved piece (E F G) from both, and laying a smear 
 of strong glue along E F G, so as to fasten them together 
 
 ?'J'J •J2, 
 
34 
 
 along tliat curved side only. Then take one sheet up by 
 the comer A or B, and let the other sheet hang freely. If 
 there be no crumpling or stretching, we shall thus obtain a 
 developable surface of two sheets. A tangent to the curve 
 E F G, such as K F L, will be partly on one sheet and 
 partly on the other. 
 
 The curve E F G which connects the two sheets is called 
 the edge of regression. It is not a very significant name, 
 being a bad translation of the French arete de rehroussement. 
 Still it is the usual name. 
 
 In the case of cones, this curve degenerates into a point 
 It then ceases to be a necessity that the angle between the 
 two sheets should be infinitesimal, and, accordingly, this 
 angle is then generally finite. 
 
 The analytical criterion of a developable smface is derived 
 from the consideration that its generating lines must be tan- 
 gents to a curve in space. In analytical language, they must 
 have an envelope. It must, therefore, be possible to get rid 
 of the parameter which distinguishes one line from another 
 by differentiating with regard to that parameter. Going back 
 to equation (2), namely, 
 
 r + 2sc2 + tc^^ = 0, 
 
 in which there is only one parameter, Co, and differentiating 
 with regard to that, we get 
 
 s -\- tC.^=zO 
 
 Now eliminating Ca by ordinary algebra, we obtain, as the 
 differential equation of ruled surfaces, 
 
 rjf-/-= 0. 
 
 The geometrical interpretation of this is that a section 
 parallel to the tangent plane at any point, and infinitely near 
 to it, is always a parabola. In other words, the curvature 
 at every point is parabolic. 
 
 To resume then, the differential equation of ruled siu'faces* 
 is the result of eliminating c between 
 
 r + 2sc + ^c^ = 
 
 and a + 3/3c + Syc^ + hc^ = 
 
 * It seems at first sight rather singular that the elimination of a single 
 constant should introduce four new partial differentials ; but it must be recol- 
 lected that the partial differentials are here only accidental. The work we are 
 doing is implicit differentiation, and the variation of the whole equation 
 simply amounts to the introduction of one new element. This consideration 
 seems to dispose (affirmatively) of the question whether the resulting equation 
 represents ruled surfaces only. The converse question, whether ruled surfaces 
 are included in it, is evident at first sight. 
 
35 
 
 while that of developable surfaces is 
 
 rt = s\ 
 
 It will be observed that this is the condition that tha first 
 equation should give two equal values for c. 
 
 Cones and Cylinders, 
 
 The differential equations previously obtained are not 
 those which distinguish these surfaces from other ruled or 
 developing surfaces. On the contrary, they are restricted by 
 conditions which settle the position of the vertex of the cone, 
 or the direction of the generators of the cylinder. They are 
 consequently expressed by differential equations of the first 
 degree. If we desire to eliminate the constants of the vertex, 
 or of direction, we must proceed to third clifferentials. 
 
 Writing the equation of the cone as 
 
 and regarding x and y as independept variables, it is easy to 
 verify that 
 
 «y - f _ a§ — h __ iSi_^■_^^ 
 r ""* 25 ■" T~ ' 
 
 but these equations are also derivable from the general 
 equation of developable surfaces, rt = s-, and do not 
 distinguish conical surfaces from all others. 
 
 Jf again we write the equation of the cylinder as 
 ny — mz = Y {mx — ly) 
 
 we find that the numerators of the above fimctions vanish, or, 
 as the I'esult may be more simply stated 
 
 — = -= - 
 
 i3 7 5 
 
 and this appears to distinguish the cylindrical from other 
 developable surfaces. 
 
 Tlie theory of this part of the subject is recent, and far 
 from complete. Meanwhile it is certain that what goes 
 before is strictly true as stated ; but the student must be 
 cautious of drawing inferences which go beyond the text. 
 For instance, Mr. Cayley has shown that the equations 
 
 do not represent cylinders only, or even ruled surfaces only. 
 It is when taken in connexion with the equation of develop- 
 able surfaces that they represent cylinders. 
 
 29992. D 
 
36 
 
 The complexity of these results was to be expected. The 
 general equation of cones involves an arbitrary function and 
 three arbitrary constants, while that of cylinders involves a 
 function and two constants. Now the ordinary practice in 
 the formation of differential equations is to consider arbi- 
 trary functions on the one hand, as yielding what is called a 
 general solution, or primitive ; and arbitrary constants on the 
 other hand, as yielding a complete solution or primitive. The 
 simultaneous elimination of functions and constants would 
 naturally be more complex, and the result at which we have 
 arrived, namely, that it leads to the existence of simultaneous 
 equations in differentials, is only reasonable. 
 
 The equations of the cones 
 
 ay — /3^ _ aS — ^7 __ /3S — y^ 
 V~ ~ ~^s - t 
 
 in virtue of the relation rt =5^, lead to the equation of the 
 third degree 
 
 4 {ay - f) (^S - y') = (.S - ^yY- 
 
 This equation is more general than that of conical surfaces. 
 Its integration has not yet been effected, but it is known to 
 be satisfied by surfaces traced by common parabolas inter- 
 secting consecutively ; moreover, it is easily shown that it is 
 also satisfied by the equation of the ruled surface with a di- 
 rector plane, a surface wliich is not generally developable. 
 This can easily be verified by taking the director plane 
 parallel to the axis of z^ when it may be wTitten as 
 
 z =. x¥ {ax -f hy) + yf {ax + by) + (p (ax + by). 
 
 The result, being invariant, will not be affected by a change 
 of co-ordinates. The question lias not yet, however, been 
 fully studied. 
 
 The Screw, and Surfaces connected with it. 
 
 If a cylinder is put into the lathe and turned with a steady 
 motion while a tool travels along in a direction parallel to the 
 axis of the cylinder, also with a steady motion, the curve 
 traced upon the cyhnder is a screuj thread, or helix, as it is 
 sometimes called. 
 
 It is the only curve in which any portion can be super- 
 posed upon any other portion so as to fit exactly. The only 
 curve, because the circle and straight line are simply extreme 
 cases of it. If the tool is still as well as steady we get a 
 
37 
 
 circle. If the cylinder is still while the tool travels we get a 
 right line. 
 
 The most convenient form of its equations is x = a cos ^, 
 2/ = aein dy z =cS. Its tangent, of com^se, touches the 
 cylinder x^ -{- i/^ = c? and makes an angle with the axis 
 
 whose trigonometrical tangent is a -* The radius of absolute 
 
 curvature is evidently the same as that of the elliptic section 
 
 of the cylinder passing through the tangent. The minoi' 
 
 semi axis of this ellipse is «, and its major semi axis s/ d^ + c^. 
 
 c? + c^. 
 
 Hence the radius of absolute curvature is * 
 
 a 
 
 If we eliminate a and d we get a surface which is indepen- 
 dent of the diameter of the cylinder, or of the inclination 
 of the particular helix got by varying the diameter of the 
 cylinder at the same time that the pitch remains constant for 
 all values of a. This is called the shew helixoid. It is the 
 underside of a common screw staircase, or the driving side of 
 a common screw propeller. A mooiing screw and the twisted 
 surfaces of a square-threaded screw are other examples. The 
 equation is 
 
 y . z 
 ^ = tan -. 
 X c 
 
 Practical reasons made it inconvenient to include the 
 common form qf this surface in this series of models. More- 
 over, the examples of it are too frequently met with in 
 practice to render its exhibition worth encountering an incon- 
 venience. There is no difficulty in representing it by strings 
 or threads; but there is a difficulty in deforming it with 
 regularity. 
 
 The skew helixoid selected for representation (model No. 31) 
 is one in which the generating line passes through the axis, 
 and is inclined to it at a constant angle instead of being 
 at right angles to it. It is the surface of what is called a 
 screw with a triangular thread. If we call ^ the angle which 
 its generating line makes with the axis, this generating line 
 will pass through the point of the helix (a cos ^, a sin ^, c 6), 
 and also the point of the axis »= c6 — a cos <p. Its equa- 
 tions are therefore 
 
 X y z— c& 
 
 cos ~ sin 6 "~ cos 9 ' 
 
38 
 
 or eliminating 6 
 
 i;/ 
 
 cos (p s/ x^ ■\- ■if' ■=. z — c tan '- + a cos f . 
 
 X 
 
 If z is made constant, that is, if we take a section perpen- 
 dicular to the axis, we get for the polar ecj[uation of the 
 section, 
 
 ;• - k - le, 
 
 which is the spiral of Archimedes. 
 
 llevei'ting to the helix, it may be remarked that its pro- 
 jections by lines parallel to any tangent are either cycloias or 
 elliptic modifications of the cycloid, and that all its projec- 
 tions (by parallel lines) are either trochoids or elliptic modi- 
 fications of the trochoids. 
 
 The tangent to the helix has for its equations 
 
 X — a cos d __ y — a sin d __ 2 — cS 
 — a sin B a Cos B c 
 
 The first pair of terms gives :i* cos Q -\- y AnB —a. Squaring 
 all the terms, and adding the numerators and denominators 
 of the first two we get 
 
 a,'2 4- y2 ^ ^2 _ 2^ (x COS 6 + 3/ sin B) _ /^ — cB \^ 
 
 whence 
 
 2 / \ ' 
 
 ^.^ 4- t/^ — a^ ±: %^( z — cb\ 
 
 this equation, with x cos ^ + ^ sin ^ = a, represents the 
 tangent for any given value of B. When z = we get 
 r^ — a^ z=z «2 e'^, which is the involute of the circle* This is 
 geometrically evident^ if we consider the helix as traced by 
 winding a paper triangle on a cylinder. 
 
 The surface swept out by the tangent is the developable 
 helixoid (model S^o. 30). Its equation is formed by ehmi- 
 n^ting B from the above equations, which may easily be done, 
 since X cos B ^ y miB z=. a gives 
 
 ay -^ X \/ x^ -^ y^ — c? 
 The heHx itself is the edge of regression of this surface 
 
 sm 2 . •> 
 
 x" + y- 
 
ay 
 
 When the surface is actually developed or flattened, the 
 helix becomes a circle whose radius is that of the absolute 
 
 c? + c2 
 curvature of the helix, that is to say, — . 
 
 The French Skew Arch. {Biais passe.) 
 
 See model No. 27. Take the line joining the centres of 
 the large settiicircles as Jc, the middle point of it for origin, 
 and the axis of z, vertical. Then if we call the radii of the 
 small circles c and their distance from the axis of x, b, their 
 eq[uations may be wi'itten as 
 
 a:=a, (j/ — by -{- z'^ z=z c- 
 x=^a, (i/-\-by-}-z^ = c^ 
 
 A plane through the axis of x, inchned to the vertical at an 
 angle (p, will he i/ = z tan ^ : substituting y cot p for z in the 
 above equations and solving for y, we find for the extremities 
 of a generating line 
 
 X = Of 1/ s= i) sin '^^ + sin <p \^ c'^ — b^ cos ^(p 
 x = — a, 1/ =z — b sin ^f + sin ^ i/ c'^ — i^^ cos ^ 
 The plan of the generating liiie is therefof-e 
 
 X _ 1/ — smp \/ c^ —b'^ sin '^<p 
 a b sin ^f 
 
 Now putting tan ^ = - we obtain for the equation of the 
 surface 
 
 If we make x ■= and y = we get for the height at the 
 middle point z = \/c^ — b'^, which shows that the arch droops 
 in the middle, as already stated. To get the middle section^ 
 make x = 0, and we have 
 
 (Z^+ 2/y= C2(^2 +^2) _52 22 
 
 or changing to polar co-ordinates 
 
 7-2 = c2 - 62 sin '^6 
 
40 
 
 an equation which shows that the curve bears a close resem 
 blance to the ellipse. 
 
 This curved surface is essentially unsymmetrical ; it is 
 evidently of the fourth degree. 
 
 The Splay Vault of Makseille. 
 
 This is a splay vault or pendentive used to connect a 
 semicircular arch on one face of a wall with a flat or segmen- 
 tal arch on the other face. It is a ruled surface, of which 
 the intrados of the two arches are two curvilinear directors, 
 while the other director is a vertical right line. It is called 
 the Arriere-voussure de Marseille; but when one of the 
 arches is flat instead of segmental, it is sometimes called 
 the Arriere-voussure de Montpellier. TIkS latter form is 
 selected for illustration here, as the investigation is somewhat 
 simpler. The student will have no difficulty in extending 
 the algebra to meet the more general case of the segmental 
 arch. 
 
 Assuming, then, that the front arch is replaced by a 
 horizontal Hne, whose height above the springing plane is 
 h, and that the inner arch is a semicircle, radius c, let the 
 distance between the two arches be h and let the vertical 
 director be taken at a distance a behind the inner arch. Then 
 if we take the vertical director for the axis of z, the equations 
 of the other two directors will be 
 
 X = a, y'^ -\- z^ = c^ 
 X = a -\- b, z = h 
 
 Take an auxiliary plane passing through the axis of z ; its 
 horizontal trace may be written as 
 
 y =: X tan f 
 it will therefore meet the horizontal director in the points 
 
 x= {a + h)i y =: {a -\- b) tan <p, z ■= h 
 
 and the semicircular director in the point 
 
 X = a, y = a tan ^ z = a/ c^ — a^ tan ^<p. 
 
 The projection of the generator on the plane of {yz) will 
 therefore be 
 
 y — {a + b) tan ^ ^ z — h 
 
 bUnf ^h - a/ c^ ^ a^tan^^ 
 
41 
 
 Eliminating tan f by means of the horizontal trace of the 
 generator, y — oc tan ^5 we obtain after a few reductions 
 
 ab -\- hz — hx\^ _c^ x^ — a^y 
 
 fab -\- bz — hxy _c^ X 
 \ a + h — X ) 
 
 It is easy to verify, by giving x the values 0, «, and a-\-b 
 successively, that this surface really contains the three 
 directors. 
 
 This surface, like the last, is of the fourth degi'ee, and in 
 its ordinary construction is unsymmetrical. It may, how- 
 ever, be rendered symmetrical in a particular case by the 
 assumption ^ = 0. If we then transfer the origin to the 
 centre of the semicircle, the equation becomes 
 
 \b — x/ \a -\- x) 
 
 In this case all sections perpendicular to x are ellipses. 
 
 In the more usual case which occurs in actual building 
 construction, instead of a horizontal right line director there 
 is in general a very flat segment of a circle. Whatever the 
 curve may be, it will in general be possible both to construct 
 the generating line at any point geometrically and to find the 
 analytical equation of the surface, but the analytical expres- 
 sions will become mther comphcated. The expedient to be 
 used in all constructions relating to ruled surfaces with a 
 director line is to take auxiliary planes passing through that 
 line. 
 
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