4 Mr. C. W. Merrifield. Sums of the Series of the [Nov. 17, 



rence of these bands amongst the Fraunhofer lines, to warrant the 

 publication of this note. We have not yet succeeded in determining 

 precisely the conditions under which the reversal can be produced at 

 will. The most complete reversals of these bands were obtained by 

 the use of the arc of a Siemens' machine, in a crucible of magnesia, fed 

 with a considerable quantity of cyanide of titanium. The photo- 

 graphs in this case show a very complete reversal of the five bands 

 near L, and of the two strong bands near N, and a less complete 

 reversal of the six bands, beginning at about wave-length 4215. No 

 other metallic cyanides have given, when introduced into the crucible, 

 any such reversal ; nor does a stream of cyanogen led in through a 

 perforated carbon produce the effect. Various other nitrogenous 

 compounds have been tried, but the only one which has given us any- 

 thing like the effect of the titanic cyanide is borate of ammonia. 

 Some photographs taken immediately after the introduction of borate 

 of ammonia show distinctly the reversal of the group of bands near 

 L. In one case when metallic magnesium had been put into the 

 crucible, the photograph shows a reversal of only that part of the 

 series which is nearest to the magnesium group, indicating that the 

 reversal is due to the bright background supplied by the expanded 

 magnesium lines. There can be little doubt that the greater stability 

 of titanic cyanide and boron nitride than of other nitrogenous com- 

 pounds, has some influence upon the result ; and the difficulty in pro- 

 ducing the reversal at will is in securing; an absorbent stratum of suffi- 

 ciently high temperature and at the same time a sufficiently luminous 

 background. The circumstances which secure the former condition 

 almost always produce in the arc a still more intense radiation of just 

 those rays which are absorbed, without that expansion of the lines 

 wbich shows out the absorption in the case of so many metallic 

 spectra. The photographs are, however, conclusive evidence that it is 

 possible to secure both conditions. 



III. " The Sums of the Series of the Reciprocals of the Prime 

 Numbers and of their Powers." By C. W. Merrifield, 

 F.R.S. Received August 8, 1881. 



Euler has shown* that it is possible to sum the series of reciprocals 

 of powers of the prime numbers, and he has calculated the values of 

 these sums for the even powers. I thought it of some interest to 

 calculate the sums for the odd powers, and to evaluate a peculiar con- 

 stant (somewhat analogous to the Eulerian constant, — 



7=0-57721 56649 01532 86060 65) 



* "Introd. in Anal. Infin.," vol. i, cap. xv, "De seriebus ex evolutione facto rum 

 ortis," pp. 221-252. 



