644 
Proceedings of the Royal Society 
downwards, even to the limit of projection in the direction of gravity; 
but the latter has the advantage of more ready reference to practice, 
and lies more to hand in the calculations. 
In the subjoined example of a table, the velocity at the summit 
is assumed to be just equal to the terminal velocity, and the details 
are given for intervals of one-tenth part of the tabular unit of time. 
For actual use these intervals must be made much smaller, and the 
tables should be arranged for values of the characteristic velocity, 
differing slightly from each other ; the former table may be regarded 
as the beginning of the series with the title V = 0. 
Since the curves extend indefinitely both ways, such tables must 
always be incomplete ; we can do no more than carry them to the 
limits of probable utility. 
The first part of the table contains the details of the rise ; thus a 
body shot off at the instant - *50, with the velocity 2T8, and at an 
Velocity at Summit, 
V = 1-00000. 
Time. 
Elevation. 
Velocity. 
Rise. 
Hor. Distance. 
•00 
o°-oo / 
1-00000 
•ooooo 
•ooooo 
-•10 
+ 5 ”25 
1-11630 
•00518 
-•10537 
-•20 
+ 10 ”12 
1-27188 
•02158 
- -22323 
-•30 
+ 14 ”17 
1-48161 
•05092 
-•35716 
-•40 
+ 17 -42 
1-77176 
•09580 
-•51262 
-•50 
+ 20 ”27 
2-18190 
•16041 
- *69840 
Time. 
Depression. 
Velocity. 
Fall. 
Hor. Distance. 
•00 
0°"00' 
1-00000 
•ooooo 
•ooooo 
+ •10 
- 6 ”00 
•91394 
•00484 
+ -09521 
+ •20 
-12 ”25 
•85219 
•01883 
+ •18217 
+ •30 
-19 ”03 
•81041 
•04127 
+ -26201 
+ •40 
-25 ”42 
•78508 
•07157 
+ •33561 
+ •50 
- 37 ”08 
•77348 
•10921 
+ -40362 
+ •60 
-38 ”38 
•77521 
•15481 
+ *46631 
+ •70 
- 44 ”48 
•79801 
•20628 
+ -52326 
elevation of 20°"27', gradually loses speed, as shown in the third 
column, and reaches the summit at the instant -00, having there a 
velocity of FOO, and being now T60 above its original level. The 
details of the descent are given in the second part of the table. On 
