TRANSACTIONS OF THE SECTIONS. 29 



distant from the lines of striction, are investigated and shown to be ultimately 

 identical with the forms of the two cones. 



The condition of intersection of successive central axes is investigated ; and ex- 

 pressions are obtained for the curvatures of the two cuspidal edges which are then 

 generated, one in the body and the other in space. 



Througliout this investigation the motion is supposed to be specified with refer- 

 ence to rectangular axes tixed in space— the specifying elements being the three 

 component velocities of translation, the three component velocities of rotation, and 

 the differential coefficients of these six velocities with respect to time. 



The latter portion of tlie paper deals with motion in two dimensions. It is 

 shown that, in the most general motion of a plane rigid figure in its own plane, 

 the locus of points which at a given instant have straight motion is a circle 

 traversing the instantaneous centre ; but one singular point on this circle is to be 

 excepted from the locus, namely the instantaneous centre itself, which, instead of 

 being (like other points on the circle) at a point of inflection of its path, is at a 

 cusp, and is moving with infinite curvature, v^'hereas all other points on the circle 

 are moving with zero curvature. This startling result is confirmed by a com- 

 parison of the cycloid witli the trochoid. Wlien a circle rolls along a straiglit 

 line, a point just within the circumference describes a trochoid having two points 

 of inflection very near together, and the short connecting arc has a total curvature 

 of nearl}' 180^ ; whereas in the case of a point on tlie circumference, these features 

 are replaced by a cusp. 



The instantaneous curvatures of the patlis traced b}' the particles, of a moving 

 figure depend on three consecutive positions only. Four consecutive positions of 

 the figure are sufficient to determine two consecutive "circles of straight motion." 

 Those two particles of the body which are situated at the intersections of these 

 two circles might at first sight be deemed to be points of double straight motion — 

 that is, to have straight motion for two consecutive instants ; but on examination 

 it turns out that one of tliese two points is not a point of straight motion at all, 

 being, in fact, the singular point above mentioned. There is therefore in general 

 only one point of double straight motion. The position of this point is investi- 

 gated in the general case of one circle rolling on another, and its connexion with 

 the subject of " apparently neutral " equilibrium of a heavy body is pointed out. 



On certain connexions between the Molecular Properties of Metals. 

 By Professor G. Eorbes. 



On the Final State of a System of Molecules in Motion subject to Forces of any 

 lind. By J. Clerk Maxwell. 



Since reading Principal Guthrie's first letter on this subject ('Nature,' May 22, 

 1873), I have thought of several ways of investigating the equilibrium of temperature 

 in a gas acted on fjy gravity. One of these is to investigate the condition of the 

 column as to density when the temperature is constant, and to show that when this 

 is fulfilled the column also fulfils the condition that there shall be no upward or 

 downward transmission of energy, or, in fact, of any other function of the masses 

 and velocities of the molecules. But a far more direct and general method was 

 suggested to me by the investigation of Dr. Ludwig Boltzmann * on the final dis- 

 tribution of energy in a finite system of elastic bodies ; and the following is a 

 sketch of tliis method as applied to the simpler case of a number of molecules so 

 great that it may be treated as infinite. 



Principal Guthrie's second letter is especially valuable as stating his case in the 

 form of distinct propositions, every one of which, except the fifth, is incontrover- 

 tible. He has himself pointed out that it is here that we ditter, and that this 

 difference may ultimately be traced to a difference in our doctrines as to the distri- 



* Studien iiber das Gleichgewicht dcr lebendigen Kraft zwischen bewegten materieUen 

 Punkten, von Dr. Ludwig Boltzninnn. Sitzb. d. Akad. d. Wisscnseh. October 8, I8fi.') 

 (Vienna). 



