14 REPORT — 1869. 



(though here C=0) ; the two curves 



j A a;"+B/4-2D j'3/+2Ea;4-2Fy=0, 



•( A„a;2+B„y^+2D„a'y + 2Eir+2Fy=0, 



are shown to osculate when V=Yo, and only then. Also if Vo become v, when we 

 take Ao=l=B^ and Do=cos 8, 



,3. 



radius of curvature to foimer : 



(-«)' 



-f V sin S 



Thirdly. When two curves have a common point a'y', the condition that they 

 may osciiilate there is expressed by an equation, (J'*V=T^Vo, where U and T are 

 known integer functions of the coefficients of the equations which are supposed quite 

 general. 



Summary of ike Thermodynamic Theory of Waves of Finite Longitudinal 

 Disturbance. By W. J. Macqugen Rankine, C.E., LL.D., F.R.S. 



This paper contains a summary accoimt of the results of a mathematical inves- 

 tigation, the details of which have been communicated to the Royal Society. It 

 relates to the laws of the propagation of finite longitudinal disturbances along a 

 cylindrical or prismatic mass of an elastic substance of any kind, solid, liquid, or 

 gaseous. The investigation is facilitated by the use of a quantity called the 7nass- 

 velocity or smnatic velocity of propagation ; that is, the mass of matter through 

 which the disturbance is propagated in unity of time along a tube of the transverse 

 area unity ; also by expressing the relative positions of transverse planes in sucha 

 tube by means of the masses of matter contained between them, instead of by their 

 distances apart. The first part of the investigation relates to the conditions under 

 which the propagation of a wave of longitudinal disturbance of permanent type is 

 possible ; and it is shown that the principal dynamical condition of permanence of 

 type is the following : 



— E =ffi^ (a constant) ; 

 as 



in which p denotes the intensity of the longitudinal elastic pressure in absolute 

 units of force per unit of area, 5 the bulkiness of the substance (that is, the volume 

 of imity of mass), and the constant m is the mass-velocity already mentioned. (That 

 pi'oposition had been previously demonstrated in a less elementary manner by IMi". 

 Earnshaw.) Then, by the aid of the thermodynamic fwicfion, it is shown what 

 conditions as to the transfer of heat between the vibrating particles must be fulfilled 

 in order that the above relation between pressure and bulkiness may subsist. Those 

 conditions are the more nearly realized, the more abrupt the changes of density 

 which constitute the disturbance ; but they cannot be absolutel}^ realized in any 

 actual substance ; whence it is concluded that absolute pennanence of tj'pe for an 

 indefinite time in waves of longitudinal disturbance is impossible, and that it is 

 most nearly approximated to when the disturbance is abrupt. 



The latter part of the investigation relates to adiabatic u-aves ; that is, waves in 

 which there is no transfer of heat from particle to particle, and the thermodynamic 

 function is constant. In this part of the investigation, the results are to a great 

 extent identical with those previously arrived at by Poisson, Stokes, Airy, and 

 especially Earnshaw ; but are obtained by methods that are comparatively of an 

 elementary kind. It is shown that in adiabatic waves there must be a change of 

 type as the wave advances; that dm-ing that change of tv'pe the greatest compression 

 and greatest rarefaction remain constant; that the compressed parts of the wave 

 gain upon the rarefied parts of the wave, and at length overtake them, converting 

 a wave which was originally one of gradual distiu'bance into one of sudden dis- 

 turbance ; and finally, that the compressed and rarefied parts of the wave by their 

 mutual interference cause the dissipation of its energj' in molecular agitation. It 

 is conjectured that this phenomenon may be the cause of the non-existence of lon- 

 gitudinal vibrations in rays of light. It is analogous to what takes place in the 

 motion of rolling waves in shallow water, when the crests overtake the troughs 

 and at length break into them. 



