The first diagram of Gregory Exhibit No. 1 enables calculation of the sagittal angle between a straight line joining Governor Connally's torso wounds and the sagittal plane of his body as a brow raising eleven degrees. During the early Z-220s the forward direction of the limousine would have made a 11-degree lateral angle with the trajectory of a bullet fired from the sniper's nest. This coincidence of the sagittal with the lateral angle requires that Connally was facing directly forward if transited by a bullet on a straight course.
Figure 1 - Facing Forward
Viewing Connally's back and chest wounds from above show that their locations in the plane of view are consistent with a bullet fired from behind and slightly to the right of the victim. The following graphic shows this situation.
Point B represents the inshoot on Connally's back and point C marks the outshoot on his chest. The straight line BC denotes the wound track made by a bullet without deflection. Line segment AB shows the incoming trajectory and the outgoing trajectory is indicated by line segment CD. The straight line AD makes an angle h with the forward direction of the limousine shown as line segment OL. Line segment BF represents a parasagittal plane of the body and makes angle s with the direction of the transiting bullet.
The appearance of Facing Forward differs from the usual diagram showing Connally's torso wounds. Normally they align the vertical with the forward direction of the limousine and show the trajectory of the bullet as upward and slightly to the left. Instead of using the direction of the limousine as a reference, Facing Forward takes the direction of the incoming bullet as its reference. So the direction of the bullet is upward and the direction of the limousine is upward and slightly to the right. I invite readers who are uncomfortable with the appearance of Facing Forward to rotate the graphic so that OL becomes vertical and observe that the angular relationships do not change.
The advantages of using the direction of the incoming bullet as a reference direction and placing the origin of the coordinate system at Connally's center of rotation becomes obvious when considering the effect of the considerable right turn of the torso upon the trajectory of a bullet that connected the inshoot and the outshoot.
Figure 2 - Turned to his Right
If Connally were facing forward then the straight line joining his wounds, BC, would have made a zero degree angle with the direction of the incoming bullet, AB. Rotation of his torso did not change the direction of the incoming bullet nor the locations of the inshoot and the outshoot. Hence the trajectory of the incoming bullet makes an angle b with the straight line joining his wounds. This angle b conveniently measures the rotation of Connally's torso from facing straight ahead since angle h equals angle s.
Researchers disagree on the magnitude of Connally's rightward rotation on any given frame of the Zapruder film. So rather than invite quibbling by assigning a disputable value to the rotation angle b, I choose to use a variable that can have any value.
A rotated torso necessitates deflection of the transiting bullet by an angle larger than the rotation angle b. This situation arises since the bullet cannot follow the path of A to B, undergo an instantaneous deflection and continue toward the outshoot along the path of B to C. Instead the bullet gradually turns toward the exit and emerges at a deflection angle larger than the rotation angle.
The medical evidence documented an elliptical inshoot and a small tunneling wound track extending to at least the lateral chest wall. Under these conditions the bullet would have continued along a straight path for some distance before the onset of deflection. This delayed deflection would have further increased the net deflection angle of the emerging bullet. So the model of a bullet that began deflecting upon entry gives the lower bound on the actual deflection angle.
Figure 3 - Curving Wound Track
Placing the origin of a new coordinate system at the inshoot simplifies analysis of the curved path of the deflecting bullet. The coordinates of the inshoot are both zero and the ordinate of the outshoot, y0, equals the abscissa of the outshoot, x0 divided by the tangent of the rotation angle b. In terms of symbols the equation y0 = x0 / tan ( b ) ensures that the curved trajectory passes through the outshoot.
Assuming that a constant force acting along the positive X' axis deflected the bullet gives a parabolic trajectory of the form x = ( y2 / x0 ) tan 2 ( b ). Taking the derivative of x with respect to y gives dx / dy = ( 2 y / x0 ) tan 2 ( b ). This derivative vanishes at y = 0 and has the value of 2 tan ( b ) at y = y0. These derivatives give the trigonometric tangents of the angles between the Y axis and the geometric tangents to the curve. So the difference between these derivatives at the outshoot and the inshoot equals the tangent of the deflection angle d. Hence tan ( d ) = 2 tan ( b ).
If the trajectory had the form of a nth power law then the tangent of the deflection angle would equal n multiplied by the tangent of the rotation angle.
The deflecting component of the force upon the bullet decreases as deflection reduces the yaw angle. However, slowing of the bullet increases the time for a diminishing deflecting force to act. So these deviations from simple parabolic motion have opposite effects upon the shape of the trajectory. Although cancellation of these deviations does not occur, the opposition of their effects diminishes their importance.
Figure 4- Broad Maximum
The excess angle, e = invtan ( 2 tan b ) - b, representing the deflection of the bullet away from a thigh that rotated with Connally's torso through an angle b has an interesting property. This angle has a maximum when b equals the inverse tangent of one divided by the square root of two. So a rotation angle of b = 35.3 degree maximizes the excess angle as 19.5 degree.
This excess angle is weakly dependent upon the rotation angle b. For a 5-degree error in a 35-degree rotation angle, the error in the excess angle is 0.35 degree. Allowing an unreasonable 10-degree error for the 35-degree rotation angle produces a barely perceptible 1.3-degree error in the excess angle. When the error of the rotation angle becomes 15 degree the excess angle has an error of less than 3 degree.
A moderate right turn of Connally's torso is devastating for a single bullet event. Assuming that the deflection to the right is possible then the course toward the inner aspect of the thigh requires a leftward deflection of angle e to undo the effect of the rightward deflection and a further leftward deflection of about 20 degree to reach the inner aspect of the lower left thigh. The wrist was between the thigh and the chest wounds. This location would have directed any deflecting component of its force upon the bullet in an upward and rightward direction. So the bullet that exited the rotated torso had no obstacle to sharply defect it leftward.
The impossibility of a bullet having inflicted Connally's torso wounds during the early Z-220s disallows the prevalent explanation of the abrupt changes seen on ZC-223 and Z-224. During this one-eighteenth second interval, the right lapel of Connally's jacket flipped leftward purportedly in response to debris that exited from the large hole in his chest. Further some have attributed the 20-degree leftward rotation and a lesser 7.5-degree forward rotation of Connally's torso to transferred momentum from the bullet. Viewing frames ZC-223 through ZC-225 without clipping reveal an alternative and plausible cause of these abrupt changes.
The brightness of the spot in line with the left breast of Governor Connally's jacket that increased between ZC-222 and ZC-223 abruptly decreased and became invisible on ZC-224 is the first strike against the jacket bulge. This abrupt change of the spot in line with the breast is evidence of alteration that fall short of proof since the source of the bright spot is not recorded on these frames. The next strike against the jacket bulge does not have this shortcoming.
Figure 5 - WC-224"
The arrows surrounding WC-224 direct our attention to the north peristyle where a person stood with a raised arm. One numerical frame earlier on WC-223 the arm of that person hung by their side.
Figure 6 - WC-223
The Warren Commission published Time/Life frames of the Zapruder film that reported an arm rotated through an angle of sixty degrees in one-eighteenth of a second without leaving a motion blur. By contrast the clapping hands of a spectator on the north infield grass that moved a few inches as opposed to more than one foot show motion blurs on WC-223 and WC-224. The absence of a motion blur of the arm on WC-223 and WC-224 required that the arm accelerated after exposure of WC-223, reached a maximum angular speed and decelerated to near rest prior to exposure of WC-224. I contend that these motions in 1/37 second between exposures are humanly impossible and constitute proof of tampering with the Zapruder film.
Recognition of the sixty-degree rotation of the spectator’s arm as proof of alteration resolves the problems of an impossible posture for a single bullet event, a change in momenta comparable to the back and to the left motion of President Kennedy and an immaculate ejection of debris from the chest that bulged the jacket without staining Connally’s white shirt.
Appendix 1 - Physics of the Rotating Arm
At t = 0 an arm accelerates from rest, reaches a maximum angular acceleration, decelerates and comes to rest after rotating through an angle of θ = π / 3 radian at time T.
Equations one through four give the particular solutions for the angular displacement, θ(t), the angular speed, ω(t), the angular acceleration, α(t), and the angular jerk, j(t) on the interval 0 < t < T.
θ(t) = 2π / T2 [ t2 / 2 - t3 / ( 3 T ) ]
ω(t) = 2π / T2 [ t - t2 / T ]
α(t) = 2π / T2 [ 1 - 2 t / T ]
j(t) = - 4π / T3
The change in orientation of the arm, Δθ, over the interval from t = T /2 and t = T is Δθ = π / 6. Hence the clearness of the arm on frame 224 shows that the angular acceleration ceased before the exposure of this frame began. This consideration enables estimation of T as one quarter of the 55-millisecond interval between frames 223 and 224. Hence the largest permissible value T becomes 11 millisecond.
Taking the weight of an adult arm as W = 8 pound gives its mass as M = 1/4 slug. The length of this arm, L, being 2 foot gives the moment of inertia, I = 1 /3 M L2 as I = 1/3 slug foot 2. The largest angular acceleration from equation three is 2 π / T2. Under this most favorable condition the angular acceleration becomes 5.19 X 104 radian / second2.
Equation five gives the peak torque, N, required to produce the filmed motion of the arm.
N = I α = ( 1/3 ) ( 5.19 X 104 ) = 1.73 X 104 slug foot 2 / second2
The peak external force acting at the free end of the arm, F, to affect this filmed motion is F = N / L. Hence F = 1.73 X 104 slug foot 2/second2 / 2 foot = 8650 pound. This required force is two orders of magnitude larger than any muscular force of a human arm.