Prof. R. S. Ball's Description of the " Cylindroid." 181 



for convertible circular processes. At that time I derived it from 

 the maxim that heat cannot of itself pass from a colder to a hotter 

 body. I afterwards* derived the same equation in a very differ- 

 ent way, namely from the law cited above, that the work which 

 can be done by heat in an alteration of the arrangement of a body 

 is proportional to the absolute temperature, in conjunction with 

 the assumption that the heat actually present in a body is dependent 

 on its temperature only, and not on the arrangement of its consti- 

 tuents. Therewith I considered the circumstance that in this 

 way we could arrive at the already otherwise proved equation a 

 main support of that law. Now the preceding analysis shows 

 how that law, and with it the second axiom of the mechanical 

 theory of heat, can be reduced to general mechanical principles. 



XXI. Description of a Model of a Conoidal Cubic Surface called 

 the "Cylindroid," which is •presented in the Theory of the Geome- 

 trical freedom of a Rigid Body, By Robert Stawell Ball, 

 A.M., Professor of Applied Mathematics and Mechanism, Royal 

 College of Science for Ireland^. 



WE become acquainted with the geometrical freedom which 

 a rigid body enjoys by ascertaining the character of all 

 the displacements which the nature of the restraints will permit 

 the body to accept. If a displacement be infinitely small, it is 

 produced by screwing the body along a certain screw. If a dis- 

 placement have finite magnitude, it is produced by an infinite 

 series of infinitely small screw displacements. For the analysis 

 of geometrical freedom we shall only consider infinitely small 

 screw displacements. This includes the initial stages of all dis- 

 placements. 



To analyze the geometrical restraints of a rigid body we pro- 

 ceed as follows. Take any line in space. Conceive this line to 

 be the axis about which screws are successively formed of every 

 pitch from — go to + co . (The pitch of a screw is the distance 

 its nut advances when turned through the angular unit.) We 

 endeavour successively to displace the body about each of these 

 screws, and record the particular screw or screws, if any, about 

 which the restraints have permitted the body to receive a dis- 

 placement. The same process is to be repeated for every other 

 line in space. If it be found that the restraints have not per- 

 mitted the body to receive any one of these displacements, then 

 the body is rigidly fixed in space. 



* Pogg. Ann. vol. cxvi. p. 73 ; Abhandlungen ilber die mechanischen 

 Warmetheorie, vol. i. p. 242. 



f Abstract of a paper read before Section A of the British Association 

 at its Meeting at Edinburgh, August 1871. Communicated by the Author. 



