1879.] on the ' Thandcrcr ' Gan Explosion. 227 



resistance of frictiou, it would be a matter of iiuliffcrcncc as regards 

 the ultimate velocity whether the shot fell perpendicularly 30,000 feet 

 or slid down a (frictionless) slope of no matter how gradual an incli- 

 nation, or whether it descended by a curved path, so long as it did, 

 between the commencement and the end of its journey, traverse 

 30,000 feet measured perpendicularly. 



To illustrate my meaning, I will ask your attention to Diagram 5. 

 Here the straight vertical line, Fig. 1, represents 30,000 feet; if the 

 projectile were to fall from A to B, it would at B attain a velocity of 

 1400 feet in a second. The inclined line. Fig. 2, represents a (friction- 

 less) slope of an angle of 30°, but having the same vertical height 

 of 30,000 feet. If the shot were to slide down this, its velocity at 

 the bottom would still be 1400 feet a second ; but it would have 

 required double the time to attain this velocity, because it would 

 during its passage through any unit of length in its descent have been 

 subjected but to one-half of the downward impulse that it would have 

 received when falling vertically. Fig. 3 shows a frictionless concave 

 curve having a vertical height as before of 30,000 feet. Again the 

 final velocity would be the 1400 feet a second ; but the time occupied, 

 while greater than that required for the vertical fall, would be less 

 than that needed to pass down the slope, because, as will be seen, 

 the early part of the downward journey is made nearly vertically, and 

 thus the shot has already attained a high speed before it reaches 

 the lower and flatter parts of the course where the downward impulse 

 of gravity and the acceleration are but small. Fig. 4 shows a 

 convex frictionless curve of the height of 30,000 feet. In this case 

 also, when the shot had reached the bottom, the velocity would be 

 the 1400 feet a second, but the time required would be more than in 

 any of the preceding cases, because the first part of the journey 

 is made upon a path which departs but gradually from the horizontal, 

 and therefore the motion of the shot is but slow while traversing 

 this first part. 



A consideration of these four figures will show, that so long as a 

 certain total impulse is applied, it is a matter of unimportance as 

 regards the velocity when produced, whether that impulse be large 

 and uniform and needing therefore to act but through a small space 

 and for a short time as in Fig. 1, whether it be less and uniform, and 

 needing therefore to act through a greater space and a longer time 

 as in Fig. 2, whether it be variable as in Fig. 3, where it is great to 

 begin with, and becomes less towards the end, a condition of things 

 requiring comparatively a short time, or whether it be variable as in 

 Fig. 4, where it is small to begin with, and becomes greater towards 

 the end, a condition demanding the longest time of all. 



Kow these propositions, which are true when a body is caused 

 by the action of gravity to attain a velocity of 1400 feet a second, 

 are equally true when that velocity arises from the body being im- 

 pelled by a force vastly superior to that due to the action of gravity, 

 and needing therefore to be exercised through a correspondingly 



