Running uphill: energy needs and Naismith's Rule

In this note we make two points, from which follows a suggestion for running uphill:(1) an athlete's energy needs in running up a slope of gradient θ at a speed V may be considered to have three additive elements: that needed to stay alive; that needed to run at a speed V on the level; that needed to lift the body at a rate V sin θ.(2) corresponding to Naismith's rule of 1 to 8, in whole numbers, for walking or running in mountainous country (1 foot of ascent is equivalent to 8 feet on the level), the rule for running on a treadmill is 1 to 3 and the rule for running on roads is 1 to 4 or 5.This result suggests that, if possible, a runner should zigzag up slopes steeper than 1 in 8 on the fells and zigzag up slopes greater than 1 in 4 or 5 on roads.