A study of Fallow deer (Dma dama) and Domestic dog (Canis familiaris) leads to identification of an aponeurosis in the back as an important elastic strain energy store in galloping. Kinetic energy lost by the body, as the forelegs end their backward swing and the hind legs end their forward swing, is stored briefly as elastic strain energy, and recovered in an elastic recoil. Thus energy is saved, making galloping the most economical gait for high speeds. Some strain energy is also stored in muscle fibres and in the vertebral column. Mechanical tests on the aponeurosis and vertebrae lead to estimates of the quantities of energy involved.