A theorem regarding the commutation of antenna rotations

Ordinarily when an antenna is rotated successively about mutually perpendicular earth-fixed axes such as axes pointing up, north, and west, the order in which the successive rotations are made cannot be reversed without obtaining a very different final antenna orientation. This paper gives an analytic proof that the same final antenna orientation can always be obtained with the rotation order reversed as with the rotations performed in correct order. To accomplish this it is necessary when reversing the order of rotation only to also reidentify the axes of rotation. The required reidentification is obtained by considering the axes as detached from the earth and permitted to turn with the antenna as would lines painted on the antenna before the rotations are begun.