28 SEC. 2. GEOMETRY. 



1O3. Trench Skew Arch (biais passe). 



The inner drum, of yellow thread, represents this surface. It 

 is a skew surface, with a right line director ; 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 through the right line director at 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 director 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 

 direct or perpendicular to the openings, and symmetrical 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 hyberbolic paraboloids. This deformation is, however, 

 very slight, and in practical work would be avoided altogether by adhering to 

 tbe principle of drawing a plane througb the director line. 



The opening of the voussoirs is usually determined by dividing the outer 

 semicircle into equal parts. 



This form of arch is inconvenient when 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. 



104. Staircase Vault for a square wall (vis St. Gilles carree). 



105. Staircase Vault. Model for exhibiting some properties 

 of this ruled surface, by showing how it is obtained from the 

 deformation of a cylinder (douelle de la vis St. Gilles carree). 



106. Cylinder with Helix and developable Helixoid. 



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 crumpling. This 

 happens because every two consecutive generating lines meet one another on 

 the helix. 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 helix to be shortened by lowering the 

 upper plate, and the cylinder can also be inclined. When oblique, however, 

 the curve which replaces the helix is not such a screw thread as can be turned 

 in the lathe. 



