TRANSACTIONS OF SECTION A. 601 



Employing the symbol N^ to designate the number of arbitrary constants involved 

 in the most general rational function ^ (z, v) possessing the orders of coincidence 

 here in question, this also represents the number of the equations of condition 

 which are dependent on the remaining ones. We, however, know expressions 

 for the total number of these equations and for the total number of the con- 

 stants 8. Making use of these expressions we immediately obtain for the number 

 of arbitrary constants 8 satisfying the identities (2) and (3) 



(4) N„ = N^ + « - i 2 ;^ (./' -1) - -^^ rf v'f, 



k 5=1 k s=i 



and this at the same time represents the number of arbitrary constants involved 

 in the most general function H (z, v), whose orders of coincidence for the various 



values z do not exceed the co/responding numbers rj*** . . . r'.'. Here we 



have replaced — o-j'^' . . . — o-'*' by the symbols Tj'** . . . t<*' . Formula (4) 



contains one statement of the complementary theorem in a restricted case. From 

 it, however, can immediately be deduced in a variety of forms the theorem in 

 the most general case. This has already been done in the author's book. The 

 principal object of the present paper is to give a method for obtaining 

 formula (4) with greater facility and elegance. 



6, The Genesis of Elliptic Functions, 

 By RopERT Russell, 



Department of General Physics. 



The following Papers were read : — 



1. Do the Radio-active Gases {Emanations) belong to the Aryou Series? 

 By Sir William Ramsay, K.C.B., F.R.S. 



The residues of the fractionation of 120 tons of liquid air were examined in 

 the chemical laboratory of University College by Professor Moore. After 

 removal of oxygen and nitrogen, argon, krypton, and xenon remained, and were 

 separated by methodical fractionation. The xenon amounted to about 300 cm.*; 

 it was methodically fractionated at —130°, and a final residue of 0'3 cm.^ was 

 obtained. The spectrum of this portion was photographed, and diflered in no 

 respect from that of xenon. It is practically certain that if this residue had 

 contained 1 per cent, of a denser gas, that gas would have been detected. It 

 follows therefore that if there is a heavier constituent in air than xenon, its 

 amount does not exceed 1-25 billionth of the whole. Now it is certain that 

 if such an element existed, it would be gaseous, and would be found in air. Its 

 non-existence implies either the absence of such elements from the periodic table 

 or their instability. As possible atomic weights for missing elements are 178, 

 216, and 260, it is rendered probable that they are respectively unstable 

 emanations — those of thorium, of radium, and of actinium. 



2. Oib the Number and Absorption of the /? Particles emitted by Radiuni. 



By W. Makowek. 



The experiments were undertaken to redetermine the number of /3 particles 

 emitted par second by radium C in equilibrium with one gram of radium. 

 Kftdium emanation w.as sealed in a small glass tube about 1 mm. diameter, the 



