8 REPORT 1871. 



indicate a state of things wliich could not have resulted under known laws from 

 any conceivable previous distribution. So far as heat is concerned, modem inves- 

 tigations have shown that a previous disti-ibution of the matter involved may, by 

 its potential energy, be capable of producing such a state of things at the moment 

 of its aggregation ; but the example is now adduced not for its bearing on heat 

 alone, but as a simple illustration of the fact that aU portions of our Science, and 

 especially that beautiful one the Dissipation of Energy, point unanimously to a 

 beginning, to a state of things incapable of being derived by present laws from any 

 conceivable previous arrangement. 



I conclude by quoting some noble words used by Stokes in his Address at Exeter, 

 words which should be stereotyped for every Meeting of this Association : — 

 " When from the phenomena of life we pass on to those of mind, we enter a region 



" still more profoimdly mysterious Science can be expected to do but little 



" to aid us here, since the instrument of research is itself the object of investigation. 

 " It can but enlighten us as to the depth of our ignorance, and lead us to look to a 

 " higher aid for that which most nearly concerns our weUbeing." 



Mathematics. 



Exhibition and Description of a Model of a Conoidal Cubic Surface called the 

 "Cylindroid," ivhich is presented in the Theory of the Geometrical Freedom of 

 a Rigid Body. By Robeki Stawell Ball, A.M., Professor of Applied 

 Mathematics and Mechanism, Royal College of Scietice for Ireland. 



We become acquainted with the geometrical freedom which a rigid body enjoys 

 by ascertaining the character of all the displacements which the natui-e of the re- 

 straints 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 displacement 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 aU displacements. 



To analyze the geometrical restraints of a rigid body we proceed as foUows : — 



Take any line in space. Conceive this line to be the axis about which screws are 

 successively formed of everj' pitch from — oo to +qo . (The pitch of a screw is the 

 distance its nut advances when turned through the angular imit.) 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 re- 

 ceive a displacement. The same process is to be repeated for every other line in 

 space. If it be found that the restraints have not permitted the body to receive 

 any one of these displacements, then the body is rigidly fixed in space. 



If, after all the screws have been tried, the body be found capable of displace- 

 ment about one screw only, the body possesses the lowest degree of freedom. 



If one screw (A) be discovered, and, the trials being continued, a second screw 

 (B) be found, the remaining tiials may be abridged by considering the information 

 which the discovery of two screws aflbrds. It is in connexion with the two screws 

 that the cyUndroid is presented. 



The body may receive any displacement about one or both of the two screws 

 A and B. 



The composition of these displacements gives a resultant which could have been 

 produced by displacement about a single screw. 



The locus of this single screw is the conoidal cubic sm-face which has been called 

 the "cylindroid " (at the suggestion of Professor Cayley). 



The equation of the cylindroid is 



s(.r2+y2)-2azi/=0. 



