ing sections, in which AB being the line of centers, the section to the right of this line is made by a plane passing through the axis of the screw, and through the line Ce, fig. 83; and the section to the left of the line of centers in fig. 84 is made by a plane passing through the line Dd, fig. 83 on one side of the axis of the screw, and parallel to the first. The effect of this is, that F is a direct section of the screw, while H is an oblique section: also, cte is the pitch circle of the wheel, and stw the pitch line of the screw, supposing it to act as a rack.

Nevertheless, according to the supposition already made, it appears that in these two sections, and in any other parallel to them within the wheel, the screw is required to act as a rack upon the teeth of the wheel. But whatever figure be given to the screw-thread, it is seen that the forms of these racks will necessarily be different in each section; for although the form of the thread is the same in all, it is cut at a different angle in each section, by which the teeth of H remote from the axis will be more prolonged and twisted in their form than those of F in the central section; and besides this, the successive racks will retire further from the center A of the wheel, as their section recedes from the axis of the screw; as shewn in the figure in which the rackteeth H are lower than in F.

Now it has been already shewn (Art. 84), that any form of tooth being assumed, the corresponding tooth may be assigned.

The forms of the teeth in the central plane E may therefore be made to suit those of F, and the forms of the teeth in G may also suit those of H; intermediate section. It is therefore

and so on for every possible to make an

endless screw whose thread shall be in contact with the entire side of the tooth, provided the figure of the wheel

teeth be different in every section. Also, since in every section two or three pairs of teeth may be in simultaneous contact, the screw may be in contact along the entire side of all these teeth.

172. The practical difficulty of making the teeth of a wheel of which the form in every parallel section shall be different, is very simply overcome by making the screw cut the teeth, thus:

An endless screw is formed of steel, exactly the same as the proposed one, and this is notched regularly across its threads so as to convert it into a cutting instrument or tap, and then properly hardened. The wheel having had its teeth roughly cut in the proposed number, is mounted in its frame, together with the cutting screw, and the latter is turned in contact with it, and pressed gradually nearer and nearer, cutting out the teeth as it proceeds, till it has formed them to correspond exactly with its thread; it is then taken out and replaced by the smooth threaded screw.

173. The endless screw falls under the case of two revolving pieces whose axes are not parallel and never meet. It communicates motion very smoothly, and is equivalent to a wheel of a single tooth, because one revolution passes one tooth of the wheel across the plane of centers; but, generally speaking, can only be employed as a driver, on account of the great obliquity of its action.

174. In a cutting engine by Hindley of York, an endless screw of a different form was introduced, which is thus described by Smeaton :-"The endless screw was applied to a wheel of about thirteen inches diameter, very stout and strong, and cut into 360 teeth. The threads of this screw were not formed upon a cylindrical surface, but upon a

solid whose sides were terminated by arches of circles. The whole length contained fifteen threads, and as every thread (on the side next the wheel) pointed towards the center thereof, the whole fifteen were in contact together, and had been so ground with the wheel, that, to my great astonishment, I found the screw would turn round with the utmost freedom, interlocked with the teeth of the wheel, and would draw the wheel round without any shake or sticking, or the least sensation of inequality*.

"The screw was cut by the rotation of the point of a tool, carried by the wheel itself, the wheel being driven by an ordinary cylindrical endless screw.”

Fig. 85 shews this form of endless screw, and fig. 86 is

85

86

an arrangement to shew the manner of cutting the spiral thread upon the solid, in which A is a wheel driven by an endless screw B, of the common form; C a toothed wheel

fixed to the axis of the endless screw and geering with another equal toothed wheel D, upon whose axis is mounted the smooth surfaced solid E, which it is desired to cut into Hindley's endless

Smeaton, p. 183, Miscellaneous Papers.

B

screw.

For this purpose a cutting tooth F is clamped to the face of the wheel A. When the handle attached to the axis BC is turned round, the wheel A and solid E will revolve with the same relative velocity as A and B, and the tooth F will trace upon the surface of the solid a thread which will correspond to the conditions. For from the very mode of its formation the section of every thread through the axis will point to the center of the wheel. The axis of E lies considerably higher than that of B, to enable the solid E to clear the wheel A.

The edges of the section of the solid through its center, exactly fit the segment of the toothed wheel, but if a section be made by a plane parallel to this, the teeth will no longer be equally divided, as they are in the common screw; and therefore this kind of screw can only be in contact with each tooth along a line corresponding to its middle section. So that the advantage of this form over the common one is not so great as appears at first sight.

175. If the inclination of the thread of a screw to the axis be very great, one or more intermediate threads may be added, as in fig. 87. In which

case the screw is said to be double, or triple, according to the number of separate spiral threads that are so placed on its surface. As every one of these threads will pass its own wheel-tooth across the line of centers, in each revo

87

lution of the screw, it follows, that as many teeth of the wheel will pass that line during one revolution of the screw as there are threads to the screw.

If we suppose the number of these threads to be considerable, for example, equal to those of the wheel-teeth,

then the screw and wheel may be made exactly alike, as in fig. 88; which may serve as an example of the disguised forms which some common arrangements may assume.

88

The old Piemont silk-mill is an example of disguised endless screws*.

89

176. In fig 89 is represented a method of communicating equal rotation by sliding contact between two axes whose directions if produced are parallel. Aa Bb are the axes, parallel in direction.

The axis Aa is furnished with a semicircular piece CAC, forming two equal branches, and terminated by

B

D

sockets bored in a direction to intersect the axis at right angles. The axis bB is provided with a similar pair of branches dbD, and the whole is so adjusted that their four sockets lie in one plane perpendicular to the axes. A cross with straight polished arms is fitted into the sockets in the manner shewn in the figure; and its arms are of a diameter that allows them to slide freely each in its own socket. If one of the axes be made to revolve, it will communicate to the other by means of this cross a rotation precisely the same as its own.

For let fig. 90 be a section through the cross transverse to the axis, and let AB be the axes, and the circles be those described by their sockets respectively.

Then if D be a socket of A, the arm of the cross which passes through it must meet the center A; and in

* Described in Encyc. Methodique, Manufactures and Arts, tom. 11. p. 31; and in Borgnis, Machines pour confectionner les étoffes, p. 160.