1903.] The Ultra-violet Spectrum of Radium. 295 



those which are associated with two surfaces are obtained up to the 

 second order inclusive. The necessary calculations are laborious. In 

 the case of the invariants, which are actually of the third order, only 

 the results are stated; they were obtained by solving fifty-seven 

 simultaneous partial differential equations. 



It is known from Lame's investigations that there are six equations 

 characteristic of the fundamental magnitudes when the independent 

 variables are the parameters of a triply-orthogonal system of surfaces. 

 Cayley proved that there are similarly six equations when the indepen- 

 dent variables are the parameters of three families of surfaces not 

 orthogonal to one another. These six equations, as formed by Cayley, 

 arise in the course of the construction of the invariants of the third 

 order. 



In the later part of the memoir, the invariants up to the second 

 order inclusive are geometrically interpreted. Those of the third 

 order have not yet been similarly interpreted ; geometrical considera- 

 tions are adduced to show that, when the significance of these invariants 

 is established, two new fundamental equations among the quantities 

 connected with a surface will be found to exist. 



" The Ultra-violet Spectrum of Badium." By Sir WILLIAM 

 CROOKES, F.B.S. Beceived August 1, 1903. 



[PLATES 1618.] 



The spectrum of radium has been examined and the wave-lengths 

 of many of its lines given by several observers, amongst whom I may 

 include Exner and Haschek,* Berndt,f Demar^ay,! and Bunge. 



Between these observers, however, there are great discrepancies, 

 lines given by one being absent in other lists, and the wave-lengths 

 even of strong lines varying between wide limits. Being in possession 

 of perhaps the purest radium hitherto employed for spectrum work, 

 I have used some of it in photographing its ultra-violet spectrum. 

 The negatives so obtained have enabled me to get measurements 

 from which the wave-lengths of the lines have been calculated with 

 an accuracy only limited by the accuracy of the iron lines used as 

 standards. 



Franz Exner and E. Haschek, ' Wien Akad. Sitzber.,' vol. 110, July, 1801 ; 

 ' Chem. News,' vol. 86, p. 247. 



f Q-. Berndt, ' Physikalische Zeitsckrift,' 2 Jahrg., No. 12 ; ' Chem. News,' 

 vol. 83, p. 77. 



J Deruarsay, ' Comptes Eendus,' vol. 129, p. 716 ; vol. 131, p. 258. 



C. Runge, ' Astrophysical Journal/ vol. 12, p. 1. 



