The following are the pitches commonly in use among mechanics :in. in. in. in. in. in. in.

1, 1, 1, 11, 2, 21, 3.

Prof. Willis considers the following to be sufficient below inch pitch :

Having, therefore, determined the proper pitch to be given to the tooth from formula 234., the nearest pitch is to be taken from the above series to that thus determined.

NOTE I.

EXPERIMENTS OF M. MORIN ON THE TRACTION OF CARRIAGES.

THE following are among the general results deduced by M. Morin from his experiments :

1. The traction is directly proportional to the load, and inversely proportional to the diameter of the wheel.

2. Upon a paved or a hard Macadamised road, the resistance is independent of the width of the tire when it exceeds from 3 to 4 inches.

3. At a walking pace the traction is the same, under the same circumstances, for carriages with springs and without them.

4. Upon hard Macadamised and upon paved roads the traction increases with the velocity; the increments of traction being directly proportional to the increments of velocity above the velocity 3.28 feet per second, or about 2 miles per hour. The equal increment of traction thus due to each equal increment of velocity is less as the road is more smooth, and the carriage less rigid or better hung.

4. Upon soft roads of earth, or sand or turf, or roads fresh and thickly graveled, the traction is independent of the velocity.

5. Upon a well-made and compact pavement of hewn stones the traction at a walking pace is not more than three-fourths of that upon the best Macadamised road under similar circumstances; at a trotting pace it is equal to it.

6. The destruction of the road is in all cases greater as the diameters of the wheels are less, and it is greater in carriages without than with springs.

NOTE K.

ON THE STRENGTH OF COLUMNS.

MR. HODGKINSON has obligingly communicated the following observations on Art. 430.:

1. The reader must be inade to understand that the rounding of the ends of the pillars is to make them moveable there, as if they turned by means of a universal joint; and the flat-ended pillars are conceived to be supported in every part of the ends by means of flat surfaces, or otherwise rendering the ends perfectly immoveable.

2. The coefficient (13) for hollow columns with rounded ends is deduced from the whole of the experiments first made, including some which were very defective on account of the difficulty experienced in the earlier attempts to cast good hollow columns so small as were wanted. The first castings were made lying on their side; and this, notwithstanding every effort, prevented the core being in the middle; some of the columns were reduced, too, in thickness, half way between the middle and the ends, and near to the ends, and this slightly reduced the strength. These causes of weakness existed much more among the pillars with rounded ends than those with flat ones; they are alluded to in the paper (Art. 47.). Had it not been for them, the coefficient (13) would, I conceive, have been equal to that for solid pillars (or 14·9).

3. The fact of long pillars with flat ends being about three times as strong as those of the same dimensions with rounded ends is, I conceive, well made out, in cast iron, wrought iron, and timber; you have, however, omitted it, being perhaps led to do it through the low value of the coefficient (13) above mentioned.

The same may be mentioned with respect to the near approach in strength of long pillars with flat ends, and those of half the length with rounded ends. It may be said that the law of the 17 power of the length would nearly indicate the latter; but this last, and the other powers 3·76 and 3.55, are only approximations, and not exactly constant, though nearly so, and I do not know whether the other equal quantities are not, with some slight modifications, physical facts.

4. The strength of pillars of similar form and of the same materials varies as the 1.865 power, or near as the square of their like linear dimensions, or as the area of their cross section.

TABLE I.

The Numerical Values of COMPLETE Elliptic Functions of the FIRst and SECOND Orders for Values of the Modulus k corresponding to each Degree of the Angle sin.-'k.

THE TABLES OF M. GARIDEL.

TABLE II.

Showing the Angle of Rupture of an Arch whose Loading is of the same Material with its Voussoirs, and whose Extrados is inclined at a given Angle to the Horizon. (See Art. 344.)

a= ratio of lengths of voussoirs to radius of intrados.

c = ratio of depth of load over crown to radius of intrados, so that c=3(1+a). (Art. 338.)