It is common for a hunter to find his quarry to be at a different altitude than he. This means that the hunter must shoot either uphill or downhill. This situation requires compensation of the zeroing of the weapon.

There is a general rule that shooter's use called the "Rifleman's Rule." It is stated as follows ( Reference 1 ):
In other words, pretend that the inclined target is at a horizontal distance equal to the slant range distance multiplied by the cosine of the inclination angle, and aim as if the target were really at that horizontal position. The rule is so well established that there are devices to assist shooters with implementing it (Reference 3) .

This paper examines this rule and its theoretical origins. Note that this paper uses the Mathsoft ODE solver. This routine has issues with units, so I have not used them. All units are implied in this paper. I have assumed the MKS system.
Important Utility Functions
Zeroing Angle
An important part of this discussion is the calibration of the weapon to hit a point at a specified distance. This operation is referred to as "zeroing." A function is developed below, called dZero, that determines angle, dq, that the bore must make with horizontal to hit a specific distance, rZero. Figure 1 illustrates this process.
Figure 1: Variables in the Zeroing Operation.
Numerical Simulation
This simulation provides a check on the theoretical results that we obtain later in the paper.
Basic Ballistic Simulation
Because there is so much algebraic manipulation here, it is useful to have a numerical simulation that will allow the results to be tested against an independent reference. Figure 2 shows the definition of variables that are used in the simulation.
Figure 2: Variables in the Uphill Trajectory Analysis.
Zeroing range
Projectile velocity
Simulation end point
Uphill angle
Weapon bore with respect to horizontal.
Uphill slope
Corrected Wikipedia Equations (For Comparison)
Theoretical projectile trajectory from the Wikipedia
This is the Wikipedia expression for the projectile slant range.
Theoretical distance from the origin to the impact point on the slope.
X coordinate of the impact point
Y coordinate of the impact point.
ODE solution
Since we are simulating a vacuum, there is no deceleration in the horizontal direction.
The only downward acceleration is due to gravity.
Initial Conditions
Figure 3: Agreement Between Theoretical and Numerical Simulations
Figure 3 shows that the theoretical and numerical models agree. However, we still have not derived the actual rule. We have only shown that we have a good model that can be used as a starting point and a test vehicle.
Numerical Example
The projectile hits the horizontal point where it should.
Equation from Reference 2
Here we see an interesting effect. The bullet will intersect the ground slightly further from the point of zero(rHill>rFlat). This is the key qualitative result of the rifleman's rule.
Derivation of Rifleman's Rule (Using a Taylor Approximation)
Small angle approximation
Taylor approximation
So the results here are close.
Equivalence of Wikipedia Solution and Reference Solution
Verify the Equation from Reference 1
I rederived this result so that I could verify that it was correct and not a typo. This derivation is more of a quality control activity than anything else, so I have hidden it.
Prove the Equivalence of the Wikipedia and Reference 1 Derivations
Showing equivalence really means proving this trigometric identity.
Because q and a have already been defined in this worksheet, I will modify their names to q1 and a1 in the following work. This will force Mathcad to treat them as variables.
The assertion is proved.
Using Mathsoft's solver routine above does not allow me to see each step of the derivation. I have manually included the step-by-step derivation below.
References/Attached Documents
Reference 1
This is the original reference work. All discussions stem from this book.
Reference 2
This is a web version of Reference 1 with regard to the "Rifleman's Rule."
Reference 3
This shows that hunting equipment is being built with the "Rifleman's Rule" in mind.
Reference 4
This is the errata for "Modern Exterior Ballistics." It is a useful reference.