1902.] and G cesium, and their Mutual Relations. 307 



The two limits in each spectrum towards which the two subordi- 

 nate series appeared to converge were determined by a slight modifi- 

 cation of Rydberg's method combined with graphical methods. These 

 were inserted in the diagrams and curves were drawn through the 

 points. In diagram (1) the curves were turned away from each other 

 and the points of bisection of the lines between the limits lay on a 

 straight line ; so also did the points of bisection of the lines between 

 the two more refrangible and corresponding doublets of the second 

 subordinate series. In diagram (2) the curves through the limits of 

 the series, when produced, intersected on the line of zero atomic mass. 

 This fact indicates that the difference between the two limits of the 

 series, while not proportional to the square of the atomic mass, is a 

 simple function of it. Rydberg, Kayser and Runge, and Rummel* have 

 each shown that the differences between the convergence points of the 

 subordinate series are approximately proportional to the squares of 

 the atomic masses. 



A diagram of the spectra and limits of the series was also drawn 

 for the three metals to scales of wave-lengths and atomic masses. 

 The more refrangible limits of the subordinate series and the more 

 refrangible members of the second series now lay on straight lines ; the 

 change in wave-length was thus proportional to the atomic mass. 



After a careful study of the facts and many computations, it was 

 found possible to calculate the subordinate series with considerable 

 accuracy by the following formulae. 



The first Subordinate Series. 



The two convergence points (nj) of this series are obtained as 



A 



follows : n = 22830 - 21*633 W ± — , where W is atomic mass and 



2 



A is the average difference between the doublets. The latter quantity, 

 as determined from the lines which are best suited for accurate 

 measurement, is for potassium 57*8, for rubidium 236*4, and for 

 caesium 547*6. These values, as shown above, are simple functions of 

 the atomic mass ; but the best method of expressing them is not yet 

 clear. This formula gives the following values for belonging to the 

 doublets of the first subordinate series: — Potassium, 21953*9 and 

 22011*7; rubidium, 20861*8 and 21098*2; and caesium 19677*2 and 

 20224*8. 



When these values are substituted for n in Rydberg's formula, 



N 



in which n = lQ*\-\ N - 109675, m = 3 . 4 . 5 . . . . and when we 



* * Proc. Eoy. Soc. Victoria,' vols. 9 and 10, 1897. 



Y 2 



