With a motive weight of 58-277lbs. the velocity was 6ft. per second

But when the elliptical bow-end was reduced to two circular segments, then at the same velocities the motive weights were, respectively, 36-642 and 402 lbs., which shows that, at low velocities, a dividing-line of the form of a circular arc meets with less resistance than an elliptical one; but that, . at a velocity of nine knots an hour, we can afford to have an elliptical dividing line without increasing the resistance, while the momentum is increased.

Numerous experiments proved that it was a very decided advantage when the water was made to divide in a vertical, instead of a horizontal direction, particularly in the stern.

Also it was found that, bodies when moved at a depth of six feet below the surface, experience less retardation than at the surface, if the friction be excluded, but the friction is very greatly increased. This is more particularly discernable if the bodies be very obtuse. A square plane, however, meets with more resistance at the depth of six feet, than at the surface. A plane of one foot square, moving at the rate of eight knots, meets with a resistance of 196.245 lbs.

Again, it is shown that when the length of the stern-end is nine times the half-breadth, then the

minus pressure, or suction, is nearly imperceptible; but the friction on this length exceeds the friction and suction together on a shorter stern end.

The increase of friction, which is due to the augmentation of velocity, is a maximum at a velocity of 11 feet per second, and afterwards diminishes. The friction on an area of 77.16 feet, at a velocity of 13-527 feet per second, being 40.17. The friction on a surface of 50 feet, at different velocities, is exhibited in the following table, showing the results of two different sets of experiments:

Velocities per sec. Friction

2 3 4 5 6 7 8 9 10 11 12 13.5 0.6142 1.338 2.323 3.56 5.039 6-757 8.709 10.89 13 3 15 931 18 779 23-54

0.7101 1-5882-795 4-357 6-237 8 442 10.97 13.81 16-96 20-424 24 195 30-576

In conclusion, it has been ascertained that the powers of the velocities, according to which the friction varies, are:

The head-pressure is always a little above the duplicate ratio of the velocity, but the ratio decreases as the velocity increases.

The suction is always less than the duplicate ratio, but this ratio increases slightly with the velocity.

The whole resistance is sometimes greater, and sometimes less than the duplicate ratio, according to the form of the body; but in most cases it is above the duplicate ratio at low velocities, and its ratio is less than the duplicate at high velocities.

CHAPTER III.

ON THE CALCULATION OF AREAS, VOLUMES, AND CENTRES OF GRAVITY.

LET ABDE be a trapezium, of which the angles A and B are right angles. It is required to find the distance of the

centre of gravity of the trapezium from the side AE.

Let the base AB=m, BD=b, AE=a.

Draw EC parallel to the base. Then AC is a rectangle, and ECD is a right-angled triangle.

and the distance of the centre of gravity of AC

That of the triangle from the same line is =m.

Then, if the distance of the centre of gravity of the

whole trapezium from AE be called x, we have

The following are Attwood's rules for finding the centre of gravity of a vessel's displacement.

To calculate approximately the area contained by any curvilinear plane figure.

and at equal distances from each other. Join ay,

yɛ, and draw aT perpendicular to yC.

Let AB or BC, &c., be called =m. BB=b, &c.

Aa= a;

Now, without any considerable error, we may suppose the portions ay, yɛ, &c., of the curve to be