TRACT 37.

OF GUNNERY.

225

pose it were required to determine what would be the re-sistance of the air against a 524-lb ball discharged with a ve-locity of 2000 feet per second of time. Now, by the 1st ofthe foregoing tables, the ball of 2 inches diameter, whenmoving with the velocity 2000, suffered a resistance of 1638ounces, or 1021b ; then, since the resistances, with the samevelocity, are as the surfaces nearly; and the surfaces are asthe squares of the diameters; and the diameters being 2 and5*6 nearly, the squares of which are 4 and 31*36, thereforeas 4: 31‘36 : : 102lb : 800lb nearly, that is, the 24lb ballwould suffer the enormous resistance of 800lb in its flight, inopposition to the direction of its motion!

And, in general, if the diameter of any proposed ball bedenoted by d, and r denote the resistance in the last tabledue to the proposed velocity of the 2 inch ball; then ^ d*rwill denote the resistance with the same velocity against theball whose diameter is-d.

PROBLEM II.

26. To assign a Rule for determining the Resistance due toany Indeterminate Velocity of a Given Ball.

This problem is very difficult to be performed near the truth,on account of the variable ratio which the resistance bears tothe velocity, increasing always more and more above that ofthe square of the velocity, at least to a certain extent; andindeed it appears that there is no single integral power what-ever of the velocity, or no expression of the velocity in oneterm only, that can be proportional to the resistances through-out. It is true indeed, that such an expression can be as-signed by means of a fractional power of the velocity, or ra-ther one whose index is a mixed number, viz, 2J^ or 2*1;* ; ” us 5700 = id 16 resistance, is a formula in one term only,which will answer to all the numbers in the last table of re-sistances very nearly, for the ball of 2 inches diameter; andconsequently, by means of the ratio of the squares of the di- 'ametersof the bails, for any other balls whatever. But thisvol. hi. a