326 Dr. C. Burton on Plane and Spherical 



the air to be either purely isothermal or purely adiabatic, and 

 thenceforward he treats the air as a frictionless and mathe- 

 matically continuous fluid, in which pressure and density are 

 connected by an invariable law. But in general the existence 

 of such a fluid is contrary to the conservation of energy ; 

 for as soon as discontinuity arises, energy will be destroyed. 



9. It may not be out of place to conclude this portion of 

 the subject by a short reference to a paper by Dr. 0. Turn- 

 lirz * . This author starts, as Biemann did, with the assump- 

 tion that the pressure is a function of the density only, the 

 law of pressure being further assumed to be the adiabatic 

 law ; and in order to avoid Biemann's error, he explicitly 

 uses the principle of energy applicable to continuous motion, 

 in place of the principle of momentum. But the foregoing 

 discussion will have made it clear, I think, that the solution 

 of the difficulty is not to be sought for in this direction. In 

 addition to the assumptions common to his own work and to 

 that of Tumlirz, Biemann uses only the principle of mass 

 and the principle of momentum ; and since by their aid alone 

 he arrives at a completely determinate motion, it follows 

 that any other motion consistent v T ith the same arbitrary 

 assumptions, and with the condition of mass, must violate 

 the condition of momentum. We have seen, in fact, that 

 there is dissipation of energy at a surface of discontinuity, so 

 that the condition of energy applicable to continuous motion 

 ceases to hold good. We are acquainted, too, wdth other 

 instances wdiere loss of continuity involves dissipation of 

 energy ; for example, there is the case of one hard body 

 rolling over another. 



As the result of his investigation, Dr. Tumlirz concludes 

 that as soon as a discontinuity is formed it immediately dis- 

 appears again, this effect being accompanied by a lengthening 

 of the wave and a more rapid advance of the disturbance, 

 In this way, therefore, he seeks to explain the increased 

 velocity of very intense sounds, such as the sounds of 

 electric sparks investigated by Mach f. But it has already been 

 pointed out [§ 3 (i.)],that when density and velocity are every- 

 where continuous functions of the coordinates, the front of a dis- 

 turbance advancing into still air must travel forward with the 

 velocity of infinitely feeble sounds. A greater velocity can 

 only ensue when the motion has become discontinuous. 



* " Ueber die Fortpflanzung ebener Luftwellen endliclier Sdrvring- 

 ungsweite/' Sitzungsb. der Wien. Akad. xcv. pp. 367-387 (1887). 



t Sitzungsb. der Wien. Akad. lxxv., lxxvii., lxxyiii. Cf. also W. 

 W. Jacques [On Sounds of Cannon"!, Amer. Journ. Sci. 3rd ser. xyii. 

 p. 116 (1879). 



