306 Royal Society : — 



vertical motions due to the addition and subtraction wheels be com- 

 bined together and made to act vertically upon a nut in one of the 

 abscissa wheels ; then the angles d, <p will satisfy the equation 



a sin (mQ + n(£) + a! sin (md— w<£) + a" sin (m'd + ri<f) . . . =a sin 0, 



which is the general equation of the rth. order. 



Therefore two bars moved respectively horizontally and vertically 

 by nuts in the wheels describing the angles and <p will trace by their 

 intersection the required curve. 



December 16. — Lieut.-General Sir Edward Sabine, K.C.B., Presi- 

 dent, in the Chair. 



The following communications were read : — 



" On the Thermodynamic Theory of Waves of Finite Longitudinal 

 Disturbance." By W. J. Macquorn Rankine, C.E., LL.D., F.R.SS. 

 Lond. and Edinb. 

 W The object of the present investigation is to determine the rela- 



tions which must exist between the laws of the elasticity and heat 

 of any substance, gaseous, liquid, or solid, and those of the wave-like 

 propagation of a finite longitudinal disturbance in that substance — in 

 other words, of a disturbance consisting in displacements of particles 

 along the direction of propagation, the velocity of displacement of the 

 particles being so great that it is not to be neglected in comparison 

 with the velocity of propagation. In particular, the investigation 

 aims at ascertaining : — in the first place, what conditions as to the 

 transfer of heat from particle to particle must be fulfilled in order 

 that a finite longitudinal disturbance may be propagated along a 

 prismatic or cylindrical mass without loss of energy or change of 

 type — the word type being used to denote the relation between the 

 extent of disturbance at a given instant of a set of particles and their 

 respective undisturbed positions ; and, secondly, according to what 

 law the type of a wave of finite longitudinal disturbance must change 

 when the substance through which it is propagated has, under the 

 circumstances of the disturbance, no appreciable power of transferring 

 heat from particle to particle, being in the condition which, in the 

 language of thermodynamics, is called adiabatic. The disturbed mat- 

 ter in these inquiries may be conceived to be contained in a straight 

 tube of uniform cross section and indefinite length, y 



The investigation is facilitated by the use of a quantity which the 

 author calls the Mass-velocity or SomaticVelocity — that is to say, the 

 mass of matter through which a disturbance is propagated in a unit 

 of time while advancing along a prism of the sectional area unity — 

 also by expressing the relative positions of a series of transverse planes 

 that travel along with a wave by means of the masses of matter con- 

 tained between them, instead of by their distances apart. 



Let such a transverse advancing plane coincide with that part of a 

 wave of longitudinal disturbance at which the pressure P and bulki- 

 ness S * are equal to those corresponding to the undisturbed condi- 



* The word bulkiness is used to denote the reciprocal of the density. 



