ON DEFLECTIVE FORCES. SI 



times, when the velocity is very great, as little as 30. But it usually 

 happens, in the operations of natural causes, that near the point where any 

 quantity is greatest or least, its variation is slower than elsewhere ; a small 

 difference, therefore, in the angle of elevation, is of little consequence to the 

 extent of the range, provided that it continue between the limits of 45 and 

 35 ; and for the same reason, the angular adjustment requires less accuracy 

 in this position than in any other, which, besides the economy of powder, 

 makes it the best elevation for practice. (Plate II. Fig. 17, 18.) 



The path of a projectile, supposed to move without resistance, is always 

 a parabola. This interesting proposition was first discovered by Galileo :* 

 it follows very readily from the doctrine of the composition of motion, com- 

 bined with the laws which that philosopher established concerning the fall 

 of heavy bodies. If from any points of a given right line, as many lines be 

 drawn, parallel to each other, and proportional to the squares of the corre- 

 sponding segments of the first line, the curve in which all their extremities 

 are found, is a parabola. Now supposing the first line to be placed in the 

 direction of the initial motion of a projectile, and parallel vertical lines to be 

 drawn through any points of it, proportional to the squares of the segments 

 which they cut off, these lines will represent the effect of gravitation, during 

 the times in which the same segments would have been described, by the 

 motion of projection alone ; consequently the projectile will always be 

 found at the extremity of the vertical line corresponding to the time 

 elapsed, and will therefore describe a parabola. (Plate II. Fig. 17, 19.) 



It is easy to show by experiment, that the path of a projectile is a para- 

 bola : if we only let a ball descend from a certain point, along a grobve, so 

 as to acquire a known velocity, we may trace on a board the parabola which 

 it will afterwards describe during its free descent ; and by placing rings at 

 different parts of the curve, we may observe that it will pass through them 

 all without striking them. (Plate II. Fig. 19.) 



In practical cases, on a large scale, where the velocity of a projectile is 

 considerable, the resistance of the atmosphere is so great as to render the 

 Galilean propositions of little or no use ; and a complete determination of 

 the path, including all the circumstances which may influence it, is attended 

 with difficulties almost insuperable. It appears from Robins's experi- 

 ments, that the resistance of the air to an iron ball of 4| inches in diameter, 

 moving at the rate of 800 feet in a second, is equal to four times its weight, 

 and that where the velocity is much greater the resistance increases far 

 more rapidly.t But what must very much diminish the probability of our 

 deriving any great practical advantage from the theory of gunnery, is an 

 observation, made also by Mr. Robins, that a ball sometimes deviates three 

 or four hundred yards laterally, without any apparent reason ; J so that we 

 cannot be absolutely certain to come within this distance of our mark in any 

 direction. The circumstance is probably owing to an accidental rotatory mo- 

 tion communicated to the ball in its passage through tKe piece, causing after- 

 wards a greater friction from the air on one side than on the other ; and it may 

 'in some measure be remedied by employing a rifle barrel, which determines 



* Dialogues on Motion, Dial. IV. 

 f Mathematical Tracts, 2 vols. 1761, i. 131. J Ibid. p. 150. 



