from our construction. 
following method 
ion now given, is 
 pehepee: are precisely similar to those 
by the simple con- 
ly as easy and as expedi- 
as any. 
w mk horizontal line HA a, Fig. 3. and 4. cut- 
ting the vertical drawn h Bin K;; let C be the 
centre of the circular arch EDA. Join AC, and draw 
cutting the verticals through A and B in the points 
and g. Also draw CD a Cd. Let p represent 
the angle of position EAB, and d the angle of direction 
DAB, which the axis of the piece makes with the 
line of position AB. Also let s be the angle EAD 
ich bey axis makes with the vertical. Let r express 
yrange AB, and f the fall FA necessary for commu- 
. ng the velocity of projection. Then the para~ 
meter of the bola at the point of projection is 
_ 4 f, = AE, and using S to express the sine. 
We have AB: DB=S, EAD: S, DAB,=S, z: S, 
DB : DA=S, DAB: S, DBA,=S,d:§, p. 
DA:AE=S, DEA: S, EDA,=S,d: S, 
» Therefore AB : AE=S, zx 8, d: S*, p. 
~Thatis rv4f=S,zxS,d:S?,p. 
_ And rxS*, p=4fxS,zxS,d. 
~ Hence are derived formule, which solve all the ques- 
tions contained in the problem. 
4fxS, 2x5, d. 
I [F. peallees to-ar- - 
_ 7X S*, p. 
iseqssy Sd. 
_7xS4,p. 
Il. §,d= ifxS,x 
’ The answers to the questions expressed in the two 
_. first cases are obtained by a single operation. In the 
__. first case, the maximum value of r, which corresponds 
. with the elevation AG, is a third proportional to AE 
_ and AD, and will be had by the analogy sin.’ p: sin. 
—bpa4fir. : 
' We also may remark, that the ranges made with the 
same velocity, and on the same declivity, are as the pro- 
ducts of the sines of d and of z. 
_ But these formula do not afford so ready an an- 
swer, when d is the thing wanted, as one would expect 
from. their simplicity. When d is unknown, z is also 
unknown. In this case we must remark, that S, z x S, d 
isequal inser 2+4 and that z+-d=p. 
ivee 
_ This changes our formula into r x sin.? p= 4f X 
ae a ae Fe) COS. Z & d— Cos. p= 2/K 
cos. z »d—2 fx cos. p. Therefore we have 
~ ; rxsin. *p42 fx cos. p=2fx cos. z » d. 
». Having obtained the arch z « d, and having z4+d=p, 
7 we easily obtain d, it being = ot hdl The process 
ig much expedited by the help of a table of natural 
sines.. We must remember, that when the projection 
is made on an ascending plane, the quantity 2f x cos. p, 
_ is to be added to rx sin.*p; but that it is to be su 
tracted from it if the projection is made on a declivity. 
‘But a plainer, m may be taken, although not 
principle. The 
ook eater | deduced from the 
position of the object B being known, its horizontal 
a are 
cy 
AAAI pp og 
ws - ua 
- : 
ee Spee 
GUNNERY. 
575 
distance AK is known, Call this 4. The middle di- Parabolic 
aon AG is Wee fOnavnte 2h JF A isalso known, Theory of 
ng =2f. Now fC=2fx tan. fAC, =2 fx cotan, p, , MMe 
Cal this & Then Celeckt hae cad, ing 
as the projection is made on an ascendin 
ing plane. Now we have 
SA: Cg=cos. p: cos.d—z, or 
 2f: h==b=cos. p : cos. d—z, 
Then to }p(=4d+4z) add } d—z, and we obtain d. 
This is, in fact, the process to which we are ultimate- 
yo by every method that is taken for the solution of 
case of the problem. - 
The construction suggests another process, which 
may be more acceptable to some readers. The angle 
SAG is 4 EAB. Therefore f G=2 fx tan. } p, and 
2f X tan. } p—AK=g G, = the versed sine f Bors 
CA, or Sp being radine. 
There are two questions more that must be solved 
before the artillerist can have all the information he 
requires. In throwing of shells, it is of liar im- 
portance that the fuse of the shell burn during the 
whole time of the flight, but no longer; and it would 
be best of all were it ended when the shell is about 
six feet from the ground, This requires an exact 
knowledge of the time of the flight. 
_ The time of the flight is the same with that of fall- 
ing through DB. We must therefore calculate DB in 
feet. 
_ DB. ae DB = rXsin, d 
Then ¢= 7 3 f= 16°? = iesina” From the 
sum of the logarithms of the range (measured in feet) 
and the sine of the direction, take the sum of the 
rithm of 16 and the sine of the zenith distance, and 
half the remainder is the time of the flight, measured. 
in seconds. 
If the best or middle direction had been chosen,, 
which is generally not far from being the case, DB is 
equal to-BA or r. Therefore in this case we have 
wail? 
t= 
Lastly, with respect to the velocity and momentum 
with which the projectile makes its stroke, this is easily 
deduced from the property of the parabolic motion. 
We know the velocity of projection, or the velocity at 
A, namely, that which is acquired by falling through 
FA. In like manner, the velocity at B is that acquired 
by falling through F «, (Ba being drawn el to. 
the horizon). Therefore, “FA; /Fa = velocity at 
A: velocity at B.. 
CHAP. It. 
On the Determination of the Initial Velocity of Projec« 
tiles from the e. ive force of Gonseter 
In our article GunrowpEr, we have already no« Robins’ 
ticed Mr Robins’ investigations respecting the explo- theory of 
sive force of gunpowder, and we have some the force of 
important corrections upen the data which he em- 8™Powdet 
ployed, We shall now proceed to give an account 
of the solution of the important problem of determining 
the initial velocity of a ball, when the elasticity of the 
powder at the instant of its firing is given, and when 
the density of the ball, the quantity of the charge, and 
