TRANSACTIONS OP THE SECTIONS. 15 



tities between the same three equations, and the two equations of any catoptrical line, 

 we get the equations to the corresponding caustic line. 



As to the application of this theory it offers no difficulty. On directing my 

 attention more particularly to surfaces of the second order, I obtained the following 

 results : — ■ 



(1) In the case of a sphere illuminated by parallel rays, the first system of catop- 

 trical lines consists of great circles passing through tbe same point, the second of small 

 circles cutting the former at right angles. The equation to the caustic surface that 

 corresponds to the first system is 



[4 (f +rj 2 + ^ ) _ a ^ 3 = 2 7a i (f +ij a ), 



a being the radius of the sphere, while the second system has for its caustic a straight 

 line passing through the centre of the sphere. 



(2) If the reflecting surface be an ellipsoid or a hyperboloid, either of one or of 

 two sheets, and the incident rays are parallel to one of the axes, the projections of the 

 catoptrical lines on the plane of the other axes are either ellipses or hyperbolas, whose 

 foci coincide with those of the section of the surface by the same plane. 



(3) In the case of an elliptic paraboloid illuminated by rays parallel to its axis, 

 the catoptrical lines form parabolas whose planes are parallel to one or the other of 

 the principal sections of the surface. The caustic surface is reduced to two parabolas 

 lying in the planes of the principal sections, and having the axis of the paraboloid for 

 their common axis, but situated in opposite directions. That which lies in the plane 

 of the greatest of the principal sections is turned in the same way as the paraboloid, 

 that lying in the perpendicular plane is turned in the opposite direction. Each of 

 these parabolas has the same focus as the principal section to which it is perpendicu- 

 lar, and a parameter equal to the difference of the parameters of the principal sections. 

 Lastly, each of these caustic lines is perpendicular to the corresponding system of 

 catoptrical lines. 



■ (4) In the case of a hyperbolic paraboloid illuminated by rays parallel to its axis, 

 the catoptrical lines also form two systems of parabolas in planes parallel to the planes 

 of the principal sections, and the caustio is again reduced to two parabolas situated in 

 the same two planes, and turned in opposite directions, each having a parameter 

 equal to the sum of the parameters of the two principal sections. 



There would be no difficulty in applying the above formulas to surfaces of revolu- 

 tion, to cylindrical conical developable surfaces, &c, but the preceding will suffice to 

 give an idea of the results that may be deduced in certain cases. 



On the Results of Bernoulli's Theory of Gases as applied to their Internal 

 Friction, their Diffusion, and their Conductivity for Heat. By Professor 

 Maxwell, F.R.S.E. 



The substance of this paper is to be found in the ' Philosophical Magazine ' for 

 January and July I860. Assuming that the elasticity of gases can be accounted 

 for by the impact of their particles against the sides of the containing vessel, the 

 laws of motion of an immense number of very small elastic particles impinging on 

 each other, are deduced from mathematical principles ; and it is shown, — 1st, that 

 the velocities of the particles vary from to oo, but that the number at any instant 

 having velocities between given limits follows a law similar in its expression to 

 that of the distribution of errors according to the theory of the " Method of least 

 squares." 2nd. That the relative velocities of particles of two different systems are 

 distributed according to a similar law, and that the mean relative velocity is the 

 square root of the sum of the squares of the two mean velocities. 3rd. That 

 the pressure is one-third of the density multiplied by the mean square of the 

 velocity. 4th. That the mean vis viva of a particle is the same in each of two 

 systems in contact, and that temperature may be represented by the vis viva of a 

 particle, so that at equal temperatures and pressures, equal volumes of different 

 gases must contain equal numbers of particles. 5th. That when layers of gas have 

 a motion of sliding over each other, particles will be projected from one layer into 

 another, and thus tend to resist the sliding motion. The amount of this will depend 

 on the average distance described by a particle between successive collisions. From 

 the coefficient of friction in air, as given by Professor Stokes, it would appear that 



