TRANSACTIONS OF THE SECTIONS. 9 



where a, b are arbitrary constants to be afterwards determined by the condi- 

 tions 



S 1 =e 1( S a =e a 



This result is not new ; it was in fact known to Euler, though he admitted that 

 his own demonstration was not complete. I have never yet been able to find in 

 other works a sufficiently rigorous demonstration of this method. 



On the Dynamical Theory of Gases. By Prof. J. C. Maxwell. 

 The phenomena of the expansion of gases by heat, and their compression by press- 

 ure, have been explained by Joule, Claussens, Herapath, &c., by the theory of their 

 particles being in a state of rapid motion, the velocity depending on the temperature. 

 These particles must not only strike against the sides of the vessel, but against each 

 other, and the calculation of their motions is therefore complicated. The author 

 has established the following results: — 1. The velocities of the particles are not 

 uniform, but vary so that they deviate from the mean value by a law well known in 

 the " method of least squares." 2. Two different sets of particles will distribute 

 their velocities, so that their vires vivas will be equal j and this leads to the chemical 

 law, that the equivalents of gases are proportional to their specific gravities. 3. From 

 Prof. Stokes's experiments on friction in air, it appears that the distance travelled by 

 a particle between consecutive collisions is about -y — th of an inch, the mean 



velocity being about 1505 feet per second; and therefore each particle makes 

 8,077,200,000 collisions per second. 4. The laws of the diffusion of gases, as 

 established by the Master of the Mint, are deduced from this theory, and the 

 absolute rate of diffusion through an opening can be calculated. The author intends 

 to apply his mathematical methods to the explanation on this hypothesis of the 

 propagation of sound, and expects some light on the mysterious question of the 

 absolute number of such particles in a given mass. 



Supplement to Newton's Method of resolving Equations. 

 By the Abbe Moigno. 

 This was a mathematical paper, showing a method of greatly shortening and 

 facilitating the finding of the roots of equations of a high order by the method of 

 limits. 



Note on the Propagation of Waves. 

 By G. Johnstone Stoney, M.A., M.R.I. A. 



This communication aimed at introducing less imperfect geometrical conceptions 

 into the study of wave propagation, than those commonly applied. Each element 

 of the front of a wave has been usually taken as the origin of a spherical disturbance, 

 and the subsequent position of the wave simply regarded as the envelope of all such 

 shells. This mode of treatment has the disadvantage of so imperfectly representing 

 the phenomena, that it leads to great embarrassments. Thus it leaves the direction 

 in which waves are propagated enveloped in great mystery, and most of the methods 

 which have been suggested by geometers for removing the obscurity have failed to 

 be satisfactory. The difficulty at once vanishes if we fix our attention in the first 

 instance on the element whose disturbance at a given moment we wish to determine, 

 and consider, along with its previous condition, all the influences which reach it at 

 that moment. A spherical shell described round this disturbed element as centre, 

 will in general (if the medium be homogeneous, &c.) pass through points from which 

 the influence had started simultaneously ; and if the entire series of such shells be 

 considered, as well as the time at which the influence from each must have been 

 thrown off to reach the common centre at the same moment, it will be easily seen, 

 that, roughly speaking, the parts of the medium behind the disturbed centre were to 

 a considerable distance in the same or nearly the same phase when they contributed 

 to its disturbance, whereas those parts in front of it were in rapidly succeeding 

 phases. From this it follows that the influences arriving from behind will have a 

 great preponderating resultant in one direction, while those arriving from the parts 



