the impinging spheres; because the surfaces of sphere M R are as the squares of their diameters; that is = D1 m -- equating these two values of the whole resisting forces we Q X १ R Di and ; and since the quantities of matter in spheres are in the conjoint ratio of their magnitudes and densities, or of the cubes of their diameters and densities; it is formly resisting substances are as the absolute strengths of the fibres of the substances directly, and the diameter and specific gravities of the spheres inversely. Q. E. D. LEMMA II. "Law of the depth to which a ball will pe netrate. The whole Spaces or Depths to which Spheres impinging differently resisting substances penetrate, are as the Squares of the initial Velocities, the Diameters and specimis fic Gravities of the Spheres directly, and the absolute "Strengths of the resisting Substances inversely: or These being premised, I now proceed to resolve the following important PROBLEM. To find a general Formula, which shall express the Charge of To find the Shot. By the last of the foregoing lemmata we have generally , the squares of the velocity and weight of the ball jointly. 2 DN rs 16002 Now to apply this in the present instance it is first necessary, that a case be known concerning the penetration of a given shot into oak substance. Such a case is presented at p. 273 of Dr. Hutton's Robins's New Principles of Gunnery. It is there asserted, that an 18 pounder cast iron ball penetrated a block of well seasoned oak (such as ships of war are generally built with) to the depth of 34 inches when fired with a velocity of 400 feet per second. Making therefore this the standard of comparison for all cases where the object is of oak substance, we shall have for the charge generally, 400* X *42 2 x 1600 x 7 or, SR nw DNT because the balls are of the same specific gravity, and the substance the same, or R≈r, and Nn; it will be Sw X ≈ 045 X D D G 2 that that is, the charge varies as the space to be penetrated and weight of the ball directly, and diameter of the ball inversely. But the charge by the problem being to produce the greatest effect possible in the destruction of the vessel; S in the above formula must always be put equal to the given thickness of the side; since it is well ascertained, that, for a shot to produce the most damage to any splintering object, such as oak; it must lose all its motion just as it ceases to be resisted by the object, which happens when the ball has forced its first hemisphere out of the farther surface of it. And the quantity of motion destroyed during the penetration of the first hemisphere of the ball into, and the exit of the same out of the object is precisely equal to what would be destroyed during the penetration of the ball through one of its radii, if the quantity of resisting surface was equal to half its entire superficies. Hence the charge in question will be Sw ⚫045 X D S being the thickness of the side of the ship; w the weight of the ball; and D its diameter. Example EXAMPLE. An enemy's ship is in sight: required the charge for the 42 pounder guns to destroy her as quickly and completely as possible, when the ships have approached near to each other. The side of the enemy's vessel, a 74, being 1 foot thick of oak timber. The diameter of a 42 pounder of cast iron being = .557 feet; we get ⚫045 X Su ZX 42 = .049 X =5'93806 lbs. 557 er, 5lbs. 15 ozs. for the weight of the charge sought. TABLE TABLE different Containing the various charges for the 12, 18, 24, 32, 36, Tables of and 42 pounder guns, for producing the greatest effect in charges for all cases of close action: the substance or object being of guns for oak materials from the thickness of 1 foot to that of 5 feet regularly ascending by 1 in the inches. different thicknesses of a ship's side. 24 32 15s. 3.957908 Tbs. 4:077844 Tbs. 4-197780 53 4910 | 5485624 | 5646338 36 42 6-231068 | 6·425789 | 6620510 6-815231 7-547924 7783797 8.019070 8255543 8-1643488*419484 | 8:674620 | 8.929756 |_ 9-0484409-331245| 9-614010 9-896775 32 36 42 8-491416 8.727289 8.963162 | 9.199035 | 9·184892 | 9-440028 | 9'695164 | 9950300 10-179540 | 10:462305 10-74507011-027335 |