198 



Lord Rayleigh. 



[Jan. 16, 



the problem. This function, as formed, is redundant (or, as Sylvester 

 would say, " crude "), as containing terms which do not appertain to 

 the enumeration, and it has been a principal object to obtain condensed 

 or reduced generating functions. It was not difficult to conjecture 

 that if the graphs have not more than two layers of nodes (i.e., not 

 more than two nodes along the axis of z, taken to be perpendicular to 

 the plane of the paper), but be otherwise unrestricted, the reduced 

 generating function is — 



(l-x)~ l (l-ar)~ 2 (l-^)~ 2 (1-z 4 )" 2 - . . . (1-a? 5 )" 2 . ...ad inf. 



Failure to establish, this happily led me to send the corresponding 

 crude generating function to Professor Forsyth. He furnished an 

 ingenious solution, which is on the point of appearing in the ' Pro- 

 ceedings of the London Mathematical Society.' 



Subsequent to this I obtained a proof, of quite a different character 

 and without reference to the crude form, of the same theorem which 

 will be found in this section. 



Conjecturally, the Gr.F.'s, when the layers are restricted in number 

 to 3, 4, &c, are — 



-x)' 1 (I-® 2 )" 2 {(l-aj 8 ) (l-a: 4 ). • >• *d ™f.}-\ 



(l-aj)" 1 (l-ar)- 2 (l-x 3 )" 3 {(l-a 4 ) (l-ar 5 ). ... ad inf.}-*, &c, 



and when the graphs are quite unrestricted — 



(1— x)~ l (l-a 2 )" 2 (1— aj 3 )" 3 (l-aj 4 )" 4 (l-* 5 )" 5 ad inf. 



Finally, a conjecture is made as to the form when the graphs are 

 restricted in all these dimensions. 



These conjectures are slowly being transformed into truths, and I 

 trust to present them as such to the Royal Society as Part 2 of this 

 memoir. 



IV< " On some Physical Properties of Argon and Helium." By 

 Lord Rayleigh, Sec. R.S. Received January 16, 1896. 



Density of Argon. 



In our original paper* are described determinations by Prof. 

 Ramsay, of the density of argon prepared with the aid of mag- 

 nesium. The volume actually weighed was 163 c.c, and the adopted 

 mean result was 19*941, referred to 2 = 16. At that time a satis- 

 factory conclusion as to the density of argon prepared by the oxygen 



* Rayleigh and Ramsay, 'Phil. Trans.,' vol. 186, A, pp. 221, 238, 1895. 



