302 Mr. S. H. Burbury. On the Application [Feb. 7, 



are immediately found by forming the " square-matrix " for F (z) and 

 so obtaining the conditions that F (z) should be a perfect square. 

 The various relations so found connecting the quantities jpi,p 2 , .... ,> OT > 

 and m 1 arbitrary quantities X 2 , . . . . , ^m, are algebraic integrals of 

 the above system of differential equations, and are all rational and 

 integral. I then apply the general theorem to the case m = 2, or the 

 case of elliptic integrals, and easily deduce the result given by Cayley 

 in his work entitled an ' Elementary Treatise on Elliptic Functions ' 

 (p. 340). 



I next apply the theory to the case of m = 3, and deduce two 

 algebraic integrals, and show how the remaining relations may be 

 found, and lastly to the case m = 4. 



The next subject treated of is the source of F (z), from which we 

 derive a differential equation which I call the fundamental equation 

 in the theory of Abelian integrals and functions, as its integral leads 

 us to a form which, when operated on by 8, leads us to a new 

 algebraic equation, which again leads to another by a second applica- 

 tion of the operator. By this method I obtain a number of interest- 

 ing results, many of which are now given for the first time, as far as 

 I am aware. 



I then define Abelian functions and, by a method of treatment 

 depending on what precedes, show that they are periodic functions 

 and determine their periods. 



We have at first sight 2m 1 independent periods, and I reduce 

 them to 2m 2 by an easy application of the foregoing theory. 



The above is a short abstract of what my paper contains, the most 

 important portions of it being (a) the determination of the algebraic 

 integrals in a rational and integral form ; (ft) the easy proof of the 

 periodicity of Abelian functions. 



I omit from this paper a discussion of the case in which the 

 number of variables exceeds m, as likely to make my communication 

 too lengthy. 



III. " On the Application of tbe Kinetic Theory to Dense 

 Gases " By S. H. BURBURY, F.R.S. Received January 12, 

 1895. 



(Abstract.) 



1. Start with Clausius' equation 



fjpV = T r +22Rr, 



in which p denotes pressure per unit of area, V volume, and T r 

 kinetic energy of relative motion. Also R is the repulsive force, r the 

 distance between the centres of two spheres, and the summation 

 includes all pairs. 



