CONCISE TABLES

FOR DETERMINING THE GREATEST HORIZONTAL RANGE OF A SHOT OR SHELL, WHEN PROJECTED IN THE AIR WITH A GIVEN VELOCITY;

TOGETHER WITH THE ELEVATION OF THE PIECE TO PRODUCE THAT RANGE.

Note.-These Tables are deduced from those given in the third volume

of Dr. Hutton's "Course of Mathematics."

PROBLEM XXI.

To find the greatest Range of a Ball or Shell, and the Elevation of the Piece to produce that Range.

RULE.

Enter Table A or B, and take out the logarithm of the terminal velocity answering to the given ball or shell, as the case may be, and also the logarithm of the corresponding altitude; then,

To the logarithm of the velocity with which the ball or shell is projected, add the logarithm of its terminal velocity; and the sum (abating 10 in the index,) will be the logarithm of the quotient of the initial velocity of the ball or shell, divided by its terminal velocity. With this logarithm, enter the second column of Table C, and in the adjoining or middle column will be found the corresponding degree of elevation to produce the greatest range; abreast of which, in the last column of the same table, will be found the logarithm of the range divided by the altitude. Now, to this logarithm add the logarithm of the altitude taken from Table A or B, as above directed; and the sum will be the logarithm of the greatest range.

Note.-If great accuracy be required, proportional parts must be taken for the excess of the given above the next less tabular numbers in Table C.

Example 1.

Let it be required to find the greatest range of a 24 lb. ball, when discharged with a velocity of 1640 feet, and the elevation of the piece to produce that range?

Log. of terminal velocity of a 24 lb. ball, Table A, = 7.381952

Abreast of 34:15, in last column of Table C, stands 0.463803

Hence the greatest range of a 24 lb. ball, when projected with a velocity of 1640 feet, is 7829 feet, which is nearly an English mile and a half; and the elevation to produce that range, is 34:15.

Example 2.

Let it be required to find the greatest range of a 13-inch shell, when projected with a velocity of 2000 feet per second, and the elevation to produce that range; the diameter of the shell being 12. 80 inches?

Log. of terminal velocity of a 13-inch shell, Table B,=7.278189
Given velocity of the shell = 2000

Answering to which, in Table C, is 34:49:

Log. 3.301030

=

Log.=0.579219

Hence the greatest range of a 13-inch shell, when projected with a velocity of 2000 feet, is 11986 feet, which is 24 miles and 106 feet; and the elevation to produce that range, is 34°49'.

*

Note. In this example, proportion is made for the excess of the given above the next less numbers in Table C.

PROBLEM XXII.

Given the Range at one Elevation; to find the Range at another Elevation.

RULE.

As the logarithmic sine of twice the first elevation, is to the logarithm of its corresponding range; so is the logarithmic sine of twice the other elevation, to the logarithm of its corresponding range,

Example 1...

If a 13-inch shell be found to range 11986 feet, when discharged at an elevation of 34:49, how far will it range when the elevation is 45 degrees; the charge of powder being the same at both elevations?

Example 2.

If a shell be found to range 4760 feet; when discharged at an elevation of 45 degrees, how far will it range when the elevation is 30:45; the charge of powder being the same at both elevations ?

Given the Elevation for one Range; to find the Elevation for another
Range.
RULE.

As the logarithm of the first range, is to the logarithmic sine of twice its corresponding elevation; so is the logarithm of any other given range, to the logarithmic sine of an arch. Now, the half of this arch will be the elevation required.

Example 1.

If a shell be found to range 11986 feet, when projected at an elevation of 34:49, at what elevation must it be discharged to strike an object at the distance of 12785 feet, with the same charge of powder?

If a shell be found to range 4760 feet when discharged at an elevation of 45, at what elevation must it be projected to strike an object at the distance of 4183 feet, with the same charge of powder ?

Given the Charge for one Range; to find the Charge for another Range.

RULE.

Since the ranges at the same elevation are nearly proportional to the charges, therefore-As the logarithm of the first range, is to the logarithm of its corresponding charge; so is the logarithm of the other range, to the logarithm of the charge corresponding thereto.

Example 1.

If, with a charge of 12 lbs. of powder, a shell range 5334 feet, what charge will be sufficient to throw it 2667 feet; the elevation being 45° in both cases?

If, with a charge of 9 lbs. of powder, a shell range 4000 feet, what charge will be sufficient to throw it 3000 feet; the elevation being 45 in both cases?