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Maximum range of pellets fired from a shotgun
Forensic Science International, 19 (1982) 223 - 230 0 Elsevier Sequoia S.A., Lausanne - Printed in The Netherlands
223
MAXIMUM RANGE OF PELLETS FIRED FROM A SHOTGUN
0. P. CHUGH Forensic Science Laboratory,
Havana,
Madhuban, Karnal (India)
(Received January 8, 1980; in revised form May 5, 1980; accepted July 27, 1981)
Summary A mathematical model to determine the maximum range of pellets fired from a shotgun has been suggested. It has been assumed that the air resistance experienced by a pellet is proportional to the square of its velocity (-pV2) where p = A/C with C = 0.061 86/N113 and A = 1/[17
696 + 2N + 10 000(0.2-d)]
N is the number of pellets/ounce and d is the diameter of the pellets in inches. A method for calculating the maximum range has been suggested, and the values obtained for Buck 00, size 8 and size 9 American pellets are very close to the reported experimental observations. The maximum ranges for other sizes of pellets have also been calculated. The angle of projection decreases with increase in velocity and increases with increase in the weight of the pellet. It varies between 26’ to 32” for common sizes of pellets and standard shotgun velocity. The maximum range in air is only 1 - 3% of the range attained by a pellet in vacuum.
Introduction How far a pellet fired from a shotgun travels, the angle at which the maximum range is attained and other connected questions are frequently put to ballistics experts in the courts of law. Answers have been given on the basis of some empirical relations [ 1 - 51 . Determination of maximum range experimentally presents many difficulties [4] , because of the low impact velocity, small size and large spread of pellets. However, some attempts have been made. Hatcher in his Note Booh [4] has quoted from Ordinance Technical Manual 9-1990 that the maximum ranges attained by standard shot loads of Buck 00, size 8 and size 9 when fired in full choke guns are 600, 230 and 210 yards, respectively (Table 1, column 6). These data have been used to check the results calculated in this paper. Two empirical relations have also been quoted in the literature: R (yd) = 2200
d
0)
224 TABLE
1
Comparative study perimental data Sr. No.
Nameof pellet
of maximum
No. of pellets/ ounce, N
Diameter of pellets,
range
MV (ft)
d
(in.)
of pellets calculated
by different
formulas
and ex-
Maximum range (yd) Experimental
From eqns. (1) and (3)
From eqns. (1) and _4 -
1
Empir- Empir- Vacuum ical ical formula formula (ii) (i)
17 696 1
2
3
4
5
6
7
8
9
10
11
1 2 3
Buck 00 Size 8 Size 9
8 410 585
0.33 0.09 0.08
1325 1200 1200
600 230 210
599.2 224.7 207.3
637 206 186.2
750 201 179
726 198 176
18174 14907 14907
and R (yd) = 1500/iV1’3
(ii)
They are in fact the same, as N and d are connected by the relation N1/3 d = constant. The values given by these relations (Table 1, columns 9 and 10) are quite different from the experimental results (column 6). Moreover, the empirical relations do not indicate the effect of variations in the initial velocity. An attempt has been made in this paper to develop a mathematical model to determine maximum range and the angle at which a gun may be fired to achieve the maximum range. It has been assumed that the resistance experienced by a pellet is --pp. The coefficient of resistance /L referred to by Burrard [ 21, Hatcher [ 41 and Jauhari and Sinha [ 61 is given by p =A/C where A = l/17 696
and c = O.O6186/N1’3 The standard cartridges, loading Buck 00, have a muzzle velocity (MV) of 1325 ft/s, while sizes 8 and 9 American shot have MVs of 1200 ft/s [7]. Using these values and that of p given above, the maximum ranges of the pellets are 637, 206 and 186.2 yd, respectively (Table 1, column 8). The gap between the theoretical calculations and the experimental results is too large. For Buck 00 the calculated range is greater, while those for sizes 8 and 9
225
pellets are smaller than the experimental results. In order to reduce the discrepancies, the constant A has been modified as: A = 1/[17 696 + 2N + 10 OOO(0.2 -d)]
By using this value of A, the maximum ranges of Buck 00, size 8 and size 9 were calculated as 599.2, 224.7 and 207.3 yd, respectively (Table 1, column 6). These values are almost the same as those obtained experimentally. The equations developed here can be programmed on a calculator or a computer, and the results can be obtained conveniently and quickly. Y
v
SIN 8
J
VO
k
&
v cos
9
-!
0
*Y
Fig. 1. Velocity components
along the axes.
Equation of motion Let V,, be the velocity of projection, (Y the angle of projection and V the velocity of a pellet at time t making an angle 19with the x axis. The components of velocity V along the x and y axes are V case and Vsine respectively (Fig. 1). Assuming that the resistance along an axis is proportional to the square of the velocity in that directions, the equations of motion in the two directions are
vdg=-/.Iv%os2e dV
V -
dy
dV
= -
dt
= -I-( p sin2 8 - g (for upward motion)
(B)
226
dV dV V - = -- = -pV2sin2!3 dy dt
+ g (for downward motion)
(0
Equation (A), on integration, gives x
=J-ln(l I*
+pV,Tcosa)
where /J =A/C
(2)
A = 1/[17 696 + 2N + 10 000(0.2 -d)]
(3)
C = 0.061 86/N1’3
(4)
Equation (B) on integration gives (5) Equation (B), also on integration, and after applying boundary conditions, gives H, the maximum height. Equation (C), on integration and simplification, gives a relation between H and T. Eliminating H between these relations we get
where (7)
P = g/a2
a is constant and represents the terminal velocity of the pellet. From eqns. (5) and (6) the total time of flight T is obtained. If P(, V, and QIare known, the range can be obtained from eqn. (1).
Maximum range Maximum range can be determined by differentiating eqn. (1) and equating dx/da to zero. We get, after simplification, dT/da = T tan 01
(3)
Differentiating eqns. (5) and (6) with respect to (Ywe have dT -= da!
v, cos cr g
1 (1 + V$sin2a/a2)“2
1 +.1+ v”, sin2a/a2
1
(9)
227
The value of cx which satisfies both eqns. (8) and (9) will give the maximum range when substituted in eqn. (1). The evaluation of cufrom these two equations may present some difficulty but it can easily be found by plotting dT/da and T tan (Y against CYgiven by eqns. (8) and (9). The common value of 01in the two functions is the one where the two curves intersect. z1 and z2 for a Buck 00 pellet have been plotted (Fig. 2). These two curves intersect at a! = 29”. Alternatively, a very close approximation of (Y can also be obtained by computing
(10) where z1 is the numerical value of T tan (Ywhen (Y= 30”, z2 is the numerical value of dT/da from eqn. (9) when cx = 30”, and T is the total time of flight when a! = 30”. The value of (Yfor maximum range is given by the equation fl’=30°
+e
(11)
where 8 is the algebraic quantity obtained from eqn. (10). Substituting the values of z1 and z2 from Table 3 in eqn. (10) we get CX’ = 29”.
21 20 I I9 18 17
16
ANGLE
OF PROJECTION
(Y
Fig. 2. Graphical method to determine OLfor maximum range for Buck 00 (N = 8, MV = 1325 ftls).
228 TABLE
2
Maximum 1070) Sr. No.
range of pellets
Name of pellet
and the angle at which
No. of pellets/ ounce, N
the maximum
range is achieved
Diameter of pellets, d (in.)
Range at cr = 30
Maximum range
WI
WI
Angle at which maximum range is achieved, (11 (deg)
1
2
3
4
5
6
7
1 2 3 4 5 6 7 8 9 10 11 12 13 14
L.G. SG Sp. SG SSG AAA BB 1 3 4 5 6 7 8 9
6 8 11 15 35 70 100 140 170 220 270 340 450 580
0.360 0.332 0.298 0.269 0.203 0.161 0.143 0.128 0.120 0.110 0.102 0.095 0.087 0.080
587.7 551.9 515.4 480.9 394.8 333.8 305.8 287.4 268.3 251.9 239.8 226.8 212.4 200.5
587.7 551.9 515.4 480.9 394.8 333.9 305.8 281.5 268.4 252.0 239.8 226.9 212.5 200.6
30.9 30.6 30.2 29.9 29.0 28.4 28.0 27.8 27.6 27.4 27.3 27.1 26.9 26.7
TABLE Ballistics
(MV =
3 of Buck
00 shots at different
angles of projection
(ft)
1
2
5 10 15 25 30
1437.5 1647.2 1732.0 1792.7 1797.2
0.5376 1.6916 3.1166 6.5357 8.5520
53.0406 29.2613 18.7455 10.1543 8.0365
35 45 60 75 85 90
(m=.) 1790.4 1747.2 1603.0 1302.8 824.2 0
10.8144 16.3897 29.9421 66.4059 204.5694
6.5231 4.4684 2.5320 1.1645 0.3793 0
MV = 1325
Range x
dT -
Angle of projection
ft/s;
terminal
velocity
T tan (Y
da 4
3
a = 127.77
ftls; N = 8.
229 TABLE Range
4 of Buck
00 shots with different
MVs at 30” angle of projection Ttana
dT da!
Angle for max. range
MV
Max. range
(ftls)
R (ft)
1
2
3
4
5
1100 1150 1200 1250 1300 1350
1676 1705 1733 1760 1785 1809
8.0479 8.1693 8.2849 8.3988 8.5010 8.6020
8.2197 8.1745 8.1323 8.0922 8.0543 8.0189
30.25 30.01 29.70 29.40 29.13 28.88
Conclusions (1) The calculated maximum ranges using the modified values of A for Buck 00, size 8 and size 9 American pellets agree with the reported experimental results (Table 1). (2) In vacuum, the maximum range depends on muzzle velocity only, but in a resisting medium it also varies with the size of the pellet. The ratio of range in a resisting medium to that in vacuum increases with increase in weight of the pellet. In the case of Buck 00, size 8 and size 9 American pellets it is 3.3%, 1.5% and 1.41%, respectively, of their corresponding range in vacuum (Table 1, columns 6 and 10). (3) Table 2 gives the range at 30”, the maximum range and corresponding angle at the maximum range being achieved. It is observed that the range at 30” is almost the same as the maximum range. This has been observed experimentally [ 1 - 5,8] . (4) Table 3 shows the ranges of Buck 00 pellets with MV 1325 ft/s at various angles of projection. The ranges have been plotted against (Yand the number of pellets per ounce. It is observed that maximum range is attained at an angle of projection of 29”. Accurate values of (Yfor various pellets are given in Table 2. The maximum range in vacuum is attained at 45”. (5) The angle of projection for maximum range decreases with increase in velocity (Table 4). (6) The value of A suggested in this paper gives a maximum range closer to the experimental values at an angle of projection around 30”. The values of A for other angles of projection to achieve proximity to the experimental results may need some further modification. Mathematical analysis of the phenomena of stringing and dispersion should also be possible. Work on these aspects is in progress.
230
Nomenclature A C d N R T T UP T Dll V VO X 21 22
constant ballistics coefficient diameter of pellet (inches ) number of pellets/ounce maximum range = Tup + TD,, total time of flight time of upward flight (s) time of downward flight (s) velocity of pellet at any time muzzle velocity of pellet (ft/s) horizontal distance covered numerical value of T tan a when (Y= 30” numerical value of dT/da! when (Y= 30”
Greek symbols a angle of projection which a pellet makes with the positive direction of
the x axis
P
P
= A/C, coefficient of resistance density of lead (lb/ft3)
Acknowledgement Thanks are due to Shri P.C. Wadhwa, I.P.S., Inspector-General of Police, Haryana for his keen interest in the work.
References B. R. Sharma, Fire Arms in Criminal Investigation, N. M. Tripathi Private Ltd., 1976, p. 86. G. Burrard, The Modern Shotgun - The Cartridge, A. S. Barnes, New York, 1964, Appendix 1. Shooter’s Bible, C. Muller and J. Olson, Small Arm Lexicon and Concise Encyclopedia, Inc., South Hackensack, NJ, 1968, pp. 216 - 240. J. S. Hatcher, Hatcher’s Note Book, Stackpole Company, Harrisburg, PA, 1966, pp. 547 - 554. J. S. Hatcher and J. Weller, Firearms Investigation, Identification and Evidence, Stackpole Company, Harrisburg, PA, 1957, Chap. 15. M. Jauhari and J. K. Sinha, Wounding effect of a spherical shot falling under gravity. J Forensic Sci., 7 (1962) 346 - 350. R. B. Boughan, Shot Gun Ballistics for Hunters, A. S. Barnes, New York, 1965, p. 134. H.M.S.O. Text Book of Small Arms, H.M.S.O., London, U.K., 1929, pp. 327 - 329.
223
MAXIMUM RANGE OF PELLETS FIRED FROM A SHOTGUN
0. P. CHUGH Forensic Science Laboratory,
Havana,
Madhuban, Karnal (India)
(Received January 8, 1980; in revised form May 5, 1980; accepted July 27, 1981)
Summary A mathematical model to determine the maximum range of pellets fired from a shotgun has been suggested. It has been assumed that the air resistance experienced by a pellet is proportional to the square of its velocity (-pV2) where p = A/C with C = 0.061 86/N113 and A = 1/[17
696 + 2N + 10 000(0.2-d)]
N is the number of pellets/ounce and d is the diameter of the pellets in inches. A method for calculating the maximum range has been suggested, and the values obtained for Buck 00, size 8 and size 9 American pellets are very close to the reported experimental observations. The maximum ranges for other sizes of pellets have also been calculated. The angle of projection decreases with increase in velocity and increases with increase in the weight of the pellet. It varies between 26’ to 32” for common sizes of pellets and standard shotgun velocity. The maximum range in air is only 1 - 3% of the range attained by a pellet in vacuum.
Introduction How far a pellet fired from a shotgun travels, the angle at which the maximum range is attained and other connected questions are frequently put to ballistics experts in the courts of law. Answers have been given on the basis of some empirical relations [ 1 - 51 . Determination of maximum range experimentally presents many difficulties [4] , because of the low impact velocity, small size and large spread of pellets. However, some attempts have been made. Hatcher in his Note Booh [4] has quoted from Ordinance Technical Manual 9-1990 that the maximum ranges attained by standard shot loads of Buck 00, size 8 and size 9 when fired in full choke guns are 600, 230 and 210 yards, respectively (Table 1, column 6). These data have been used to check the results calculated in this paper. Two empirical relations have also been quoted in the literature: R (yd) = 2200
d
0)
224 TABLE
1
Comparative study perimental data Sr. No.
Nameof pellet
of maximum
No. of pellets/ ounce, N
Diameter of pellets,
range
MV (ft)
d
(in.)
of pellets calculated
by different
formulas
and ex-
Maximum range (yd) Experimental
From eqns. (1) and (3)
From eqns. (1) and _4 -
1
Empir- Empir- Vacuum ical ical formula formula (ii) (i)
17 696 1
2
3
4
5
6
7
8
9
10
11
1 2 3
Buck 00 Size 8 Size 9
8 410 585
0.33 0.09 0.08
1325 1200 1200
600 230 210
599.2 224.7 207.3
637 206 186.2
750 201 179
726 198 176
18174 14907 14907
and R (yd) = 1500/iV1’3
(ii)
They are in fact the same, as N and d are connected by the relation N1/3 d = constant. The values given by these relations (Table 1, columns 9 and 10) are quite different from the experimental results (column 6). Moreover, the empirical relations do not indicate the effect of variations in the initial velocity. An attempt has been made in this paper to develop a mathematical model to determine maximum range and the angle at which a gun may be fired to achieve the maximum range. It has been assumed that the resistance experienced by a pellet is --pp. The coefficient of resistance /L referred to by Burrard [ 21, Hatcher [ 41 and Jauhari and Sinha [ 61 is given by p =A/C where A = l/17 696
and c = O.O6186/N1’3 The standard cartridges, loading Buck 00, have a muzzle velocity (MV) of 1325 ft/s, while sizes 8 and 9 American shot have MVs of 1200 ft/s [7]. Using these values and that of p given above, the maximum ranges of the pellets are 637, 206 and 186.2 yd, respectively (Table 1, column 8). The gap between the theoretical calculations and the experimental results is too large. For Buck 00 the calculated range is greater, while those for sizes 8 and 9
225
pellets are smaller than the experimental results. In order to reduce the discrepancies, the constant A has been modified as: A = 1/[17 696 + 2N + 10 OOO(0.2 -d)]
By using this value of A, the maximum ranges of Buck 00, size 8 and size 9 were calculated as 599.2, 224.7 and 207.3 yd, respectively (Table 1, column 6). These values are almost the same as those obtained experimentally. The equations developed here can be programmed on a calculator or a computer, and the results can be obtained conveniently and quickly. Y
v
SIN 8
J
VO
k
&
v cos
9
-!
0
*Y
Fig. 1. Velocity components
along the axes.
Equation of motion Let V,, be the velocity of projection, (Y the angle of projection and V the velocity of a pellet at time t making an angle 19with the x axis. The components of velocity V along the x and y axes are V case and Vsine respectively (Fig. 1). Assuming that the resistance along an axis is proportional to the square of the velocity in that directions, the equations of motion in the two directions are
vdg=-/.Iv%os2e dV
V -
dy
dV
= -
dt
= -I-( p sin2 8 - g (for upward motion)
(B)
226
dV dV V - = -- = -pV2sin2!3 dy dt
+ g (for downward motion)
(0
Equation (A), on integration, gives x
=J-ln(l I*
+pV,Tcosa)
where /J =A/C
(2)
A = 1/[17 696 + 2N + 10 000(0.2 -d)]
(3)
C = 0.061 86/N1’3
(4)
Equation (B) on integration gives (5) Equation (B), also on integration, and after applying boundary conditions, gives H, the maximum height. Equation (C), on integration and simplification, gives a relation between H and T. Eliminating H between these relations we get
where (7)
P = g/a2
a is constant and represents the terminal velocity of the pellet. From eqns. (5) and (6) the total time of flight T is obtained. If P(, V, and QIare known, the range can be obtained from eqn. (1).
Maximum range Maximum range can be determined by differentiating eqn. (1) and equating dx/da to zero. We get, after simplification, dT/da = T tan 01
(3)
Differentiating eqns. (5) and (6) with respect to (Ywe have dT -= da!
v, cos cr g
1 (1 + V$sin2a/a2)“2
1 +.1+ v”, sin2a/a2
1
(9)
227
The value of cx which satisfies both eqns. (8) and (9) will give the maximum range when substituted in eqn. (1). The evaluation of cufrom these two equations may present some difficulty but it can easily be found by plotting dT/da and T tan (Y against CYgiven by eqns. (8) and (9). The common value of 01in the two functions is the one where the two curves intersect. z1 and z2 for a Buck 00 pellet have been plotted (Fig. 2). These two curves intersect at a! = 29”. Alternatively, a very close approximation of (Y can also be obtained by computing
(10) where z1 is the numerical value of T tan (Ywhen (Y= 30”, z2 is the numerical value of dT/da from eqn. (9) when cx = 30”, and T is the total time of flight when a! = 30”. The value of (Yfor maximum range is given by the equation fl’=30°
+e
(11)
where 8 is the algebraic quantity obtained from eqn. (10). Substituting the values of z1 and z2 from Table 3 in eqn. (10) we get CX’ = 29”.
21 20 I I9 18 17
16
ANGLE
OF PROJECTION
(Y
Fig. 2. Graphical method to determine OLfor maximum range for Buck 00 (N = 8, MV = 1325 ftls).
228 TABLE
2
Maximum 1070) Sr. No.
range of pellets
Name of pellet
and the angle at which
No. of pellets/ ounce, N
the maximum
range is achieved
Diameter of pellets, d (in.)
Range at cr = 30
Maximum range
WI
WI
Angle at which maximum range is achieved, (11 (deg)
1
2
3
4
5
6
7
1 2 3 4 5 6 7 8 9 10 11 12 13 14
L.G. SG Sp. SG SSG AAA BB 1 3 4 5 6 7 8 9
6 8 11 15 35 70 100 140 170 220 270 340 450 580
0.360 0.332 0.298 0.269 0.203 0.161 0.143 0.128 0.120 0.110 0.102 0.095 0.087 0.080
587.7 551.9 515.4 480.9 394.8 333.8 305.8 287.4 268.3 251.9 239.8 226.8 212.4 200.5
587.7 551.9 515.4 480.9 394.8 333.9 305.8 281.5 268.4 252.0 239.8 226.9 212.5 200.6
30.9 30.6 30.2 29.9 29.0 28.4 28.0 27.8 27.6 27.4 27.3 27.1 26.9 26.7
TABLE Ballistics
(MV =
3 of Buck
00 shots at different
angles of projection
(ft)
1
2
5 10 15 25 30
1437.5 1647.2 1732.0 1792.7 1797.2
0.5376 1.6916 3.1166 6.5357 8.5520
53.0406 29.2613 18.7455 10.1543 8.0365
35 45 60 75 85 90
(m=.) 1790.4 1747.2 1603.0 1302.8 824.2 0
10.8144 16.3897 29.9421 66.4059 204.5694
6.5231 4.4684 2.5320 1.1645 0.3793 0
MV = 1325
Range x
dT -
Angle of projection
ft/s;
terminal
velocity
T tan (Y
da 4
3
a = 127.77
ftls; N = 8.
229 TABLE Range
4 of Buck
00 shots with different
MVs at 30” angle of projection Ttana
dT da!
Angle for max. range
MV
Max. range
(ftls)
R (ft)
1
2
3
4
5
1100 1150 1200 1250 1300 1350
1676 1705 1733 1760 1785 1809
8.0479 8.1693 8.2849 8.3988 8.5010 8.6020
8.2197 8.1745 8.1323 8.0922 8.0543 8.0189
30.25 30.01 29.70 29.40 29.13 28.88
Conclusions (1) The calculated maximum ranges using the modified values of A for Buck 00, size 8 and size 9 American pellets agree with the reported experimental results (Table 1). (2) In vacuum, the maximum range depends on muzzle velocity only, but in a resisting medium it also varies with the size of the pellet. The ratio of range in a resisting medium to that in vacuum increases with increase in weight of the pellet. In the case of Buck 00, size 8 and size 9 American pellets it is 3.3%, 1.5% and 1.41%, respectively, of their corresponding range in vacuum (Table 1, columns 6 and 10). (3) Table 2 gives the range at 30”, the maximum range and corresponding angle at the maximum range being achieved. It is observed that the range at 30” is almost the same as the maximum range. This has been observed experimentally [ 1 - 5,8] . (4) Table 3 shows the ranges of Buck 00 pellets with MV 1325 ft/s at various angles of projection. The ranges have been plotted against (Yand the number of pellets per ounce. It is observed that maximum range is attained at an angle of projection of 29”. Accurate values of (Yfor various pellets are given in Table 2. The maximum range in vacuum is attained at 45”. (5) The angle of projection for maximum range decreases with increase in velocity (Table 4). (6) The value of A suggested in this paper gives a maximum range closer to the experimental values at an angle of projection around 30”. The values of A for other angles of projection to achieve proximity to the experimental results may need some further modification. Mathematical analysis of the phenomena of stringing and dispersion should also be possible. Work on these aspects is in progress.
230
Nomenclature A C d N R T T UP T Dll V VO X 21 22
constant ballistics coefficient diameter of pellet (inches ) number of pellets/ounce maximum range = Tup + TD,, total time of flight time of upward flight (s) time of downward flight (s) velocity of pellet at any time muzzle velocity of pellet (ft/s) horizontal distance covered numerical value of T tan a when (Y= 30” numerical value of dT/da! when (Y= 30”
Greek symbols a angle of projection which a pellet makes with the positive direction of
the x axis
P
P
= A/C, coefficient of resistance density of lead (lb/ft3)
Acknowledgement Thanks are due to Shri P.C. Wadhwa, I.P.S., Inspector-General of Police, Haryana for his keen interest in the work.
References B. R. Sharma, Fire Arms in Criminal Investigation, N. M. Tripathi Private Ltd., 1976, p. 86. G. Burrard, The Modern Shotgun - The Cartridge, A. S. Barnes, New York, 1964, Appendix 1. Shooter’s Bible, C. Muller and J. Olson, Small Arm Lexicon and Concise Encyclopedia, Inc., South Hackensack, NJ, 1968, pp. 216 - 240. J. S. Hatcher, Hatcher’s Note Book, Stackpole Company, Harrisburg, PA, 1966, pp. 547 - 554. J. S. Hatcher and J. Weller, Firearms Investigation, Identification and Evidence, Stackpole Company, Harrisburg, PA, 1957, Chap. 15. M. Jauhari and J. K. Sinha, Wounding effect of a spherical shot falling under gravity. J Forensic Sci., 7 (1962) 346 - 350. R. B. Boughan, Shot Gun Ballistics for Hunters, A. S. Barnes, New York, 1965, p. 134. H.M.S.O. Text Book of Small Arms, H.M.S.O., London, U.K., 1929, pp. 327 - 329.