Ranges of 
Projectiles 
in a resist. 
—_—— 
582 
The following examples will make this proposition 
easily understood. An 18 pounder, with a five inch 
iron shot, and a‘charge of 2 lb. of powder, ranged 975 
yards, at an elevation of 3° 30’. In this case, F =d 
X 300 = 1500, and ae = 0.65, which gives in the 
2d column of the Table 0.817, consequently 0.817F= 
1225 yards equal the potential range required, which 
increased in the ratio of the radius to the sine of twice 
the angle of elevation, gives 10050, for the potential 
random. The original velocity is then 
war P= 984 feet in 1”, where p is the po- 
o.= 
tential random, or 10050 yards, or 30150 feet. 
{n order to find the actual range from the potential 
range at a small angle, enter the second column of the 
Table with the quotient of the ntial range divided 
by its correspondent F, there will be found opposite to 
it in the first column, a number which, multiplied by 
F, will give the actual range required. ’ 
If the potential random deduced from the potential 
range exceeds 13,000 yards, then it ought to be correct- 
ed for the treble resistance already mentioned. In 
order to find the correct potential random, take a 4th 
continued proportional to 130,000, and the potential 
random is found by this proposition ; and this 4th pro- 
portional is the potential random, corrected for the 
treble resistance. In like manner, when the true po- 
’ tential random is given greater than 13,000 yards, we 
must take two mean proportionals between 15,000 and 
this random, and the first of these proportionals must 
be assumed, instead of the random given in all the ope- 
rations described under this proposition. 
For example, a 24 pounder charged with 12 Ibs. of 
powder, ranged about 2500 yards at an angle of 7° 15’. In 
this case, F is 1700, and a0 = 1.47, opposite which, in 
the 2d column, is 2556, which gives the potential range 
4350 yards, and the potential random 174,000 ; but as 
that is more than 13,000, we must take a 4th continued 
proportional to 13,000 and 17,400, which is $1,000 
yards, the correct potential random required, whence the 
velocity is nearly 1780 feet in a second. 
Prop. II. 
The actual range of a given shell or bullet, at an 
angle not exceeding 45°, being given, to determine its 
potential range at the same angle, and thence its po- 
tential random and original velocity. 
Let A be the angle of elevation, then multiply F by 
chai = and the product will be'E corrected for the 
given angle. Use this corrected value of F instead of 
E in the way described in Prop. I. and the potential 
range will be had, eonsequently the potential random 
and original velocity. 
Thus a mortar charged with 30 Ib. of powder, 
throws a shell 12? inches diameter, and 231 lb. weight, 
to the distance of 3450 yards, or two miles, at an 
elevation of 45°. The value of F corres onding to 
this shell is 122x300, or 3825 yards ; ie as the 
shell is only four-fifths of a solid globe, the true 
y L. 4 
value of F will axe = 8060, which when mul. 
ss 8A. ; 
tiplied by cos. 4— gives 2544 for F corrected. Now 
the quotient of the potential range divided by E; or 
GUNNERY. 
pad is 1.884, which when sought in the ‘Ist column Pr 
of the Table gives 2,280, which being iplied | 
corrected F, gives 5600 ards for the Sivonen ae 
required, This is also the potential random, ar the — 
elevation is 45°, and the original velocity of the shell 
will be paler sera Re - + 
In order to fin actual range from the potential 
range, divide corrected F by the potential range ; and 
entering the 2d column of the Table with this quo- 
tient, the number in the Ist column multiplied into cor- 
rected F will be the actual range sought. 
Prop. III. 
The potential random of a given shell or bullet being 
given to determine its actual range at45°. 
Make g=the given potential random divided by F 
corre: apis Soe the shell or bullet. © 
d=the difference between the tabular loga- 
rithms of 25 and of 9, the logarithm 10 
__ being supposed unity, Then the actual 
range sought will be 
d 
3,4F4+-2dF— 10 F when q exceeds 25, and 
. ( 
8,4 F—2 ar F when q is less than 25, 
Tn this solution g may be any number not less than 3, 
nor greater than 2500. oY 
The following Table computed. in this i shews 
the relation between the potential randoms and the ac- 
tual range at 45°, within the limits of the proposition. 
Potential Actual Range —- Potential ~— Actual Range 
Randoms, at 45°, Rardoms. at 459 
cS eee ot 5OF .. 40F 
GE sie Qadak 100- Fins 6. #65 > 
10F . 2.6F 200F . . 51F 
20.F . is 82 F 500F .. .. 588 
30 F . 86F 1000F .. 64F © 
40F .. 38F 2500F . . 70F 
It is singular to observe that so enormous. are the 
effects of a: air’s resistance, that when the potential 
random increases from 3 F to 2600 F, the actual range 
increases only from 14 F to 7F. 
In order to examine the justness of the approxima- 
tions laid down in Prop. II. and III. Mr Robins has cal- 
culated a table of the actual ranges at 45° of'a projectile, 
resisted as the square of its velocity. 1) stay 
- 
Potential Actual Range — Potential = Actual Range 
Randoms. at 45° Randoms. at 46°, 
1 F . 0963 F 65,(B 860 Bt 
25 F .- 2282 F 7.0 FP. ..s,.:2:287 1By ak 
Pee sy 4203 F 7.5F . 2.300 F 
75 F . ..6868 F $0F . 2.3859 F 
1.0.F .. 78233 F 85 F . 2414 F 
125 F . .g60 F 90F . 2467 F 
15 F 978 F 9.5 F 2.511 F 
1.75 F ..., 1.0838 oF 10.0 F . 2564 F 
2:0 F ... 1179) 110 F .. 2.654F) 5 
25 F.. 1349 F 13.0 F . 2.804 F 
3.0 F. . 1495 F 150 F . 2.937 F 
$5 F.. 1694 F 200 F . 8196 F 
40 F. 1.788 F 250 F . 3.396F ~ 
4.5 F . 1.840 F 30.0 F . 3,557 F 
5.0 F . 1930 F 400F . 3.809 F° 
55 F. 2015 500 F . 3,998 F 
60 F . 2097 F 
