1916] ON THE DISCHARGE OF GASES UNDER HIGH PRESSURES 413 Again, if Sp be the variable part of the pressure at the axis (r = 0), where p is the average density in the jet and Sw the variable part of the component velocity parallel to z. Accordingly ; ........................... (22) BR __ /„' (2-405) V( W*/cS- 1) " ................... ( } In (23) we may substitute for /3 its value, viz. and for J"0' (2'405) we have from the tables of Bessel's functions -0-5191, so that = -0-2158 .£ (a~« -F-) ...................... (24) ~~ As was to be expected, the greatest swelling is to be found where the pressure at the axis is least. A complete theory of the effects observed by Mach and Emden would involve a calculation of the optical retardation along every ray which traverses the jet. For the jet of circular section this seems scarcely practicable ; but for the jet in two dimensions the conditions are simpler and it may be worth while briefly to consider this case. As before, we may denote the general thickness of the two-dimensional jet by b, and take b + rj to represent the actual thickness at the place (z) where the retardation is to be determined. The retardation is then sufficiently represented by A, where A= (p-pi)dy- pdy-faQ + v), ......... (25) Jo Jo p being the density in the jet and /o3 that of the surrounding gas, The total stream Swdy; o o o and this is constant along the jet. Thus A-O-iprt-^j^awdy, .............. . ...... (26) G being a constant, and squares of small quantities being omitted. In analogy with (9), we may here take ............... (27)e jet of circular section, we may establish the connexion between the variable pressure along the axis and the amount of the swellings observed to take place between the disks. From (9)