Let a disturbance once set going travel along unchanged in form from A to B with velocity U.
For a single quantic of the first order (ab) is the symbol of a function of the coefficients which vanishes identically; thus (ab) =a1b2-a2bl= aw l -a1ao=0 and, indeed, from a remark made above we see that (ab) remains unchanged by interchange of a and b; but (ab), = -(ba), and these two facts necessitate (ab) = o.
We shall regard the external force as applied in the form of a pressure X per square centimetre parallel to the line of propagation and varied from point to point as required in order to make the disturbance travel on unchanged in form with the specified velocity U.
The arrangements thus established by Augustus continued almost unchanged till the time of Constantine, and formed the basis of all subsequent administrative divisions until the fall of the Western empire.
(g) Our knowledge of primitive forms of sacrifice is meagre; even were it more extensive, it would probably be impossible to determine the origin or origins of sacrifice; for no ritual has necessarily survived unchanged in form and meaning since its inception, and even permanence of form cannot be taken to imply a corresponding permanence of meaning for the worshippers.