and we easily find that there is an interesting relation between hypergeometrics whose arguments x and y are connected by the relation = If we write and therefore k+1 2 (y/4) 2 F(k+1, 1k+1; k+3; y). (15) this becomes, in terms of §, e~ (k+1)§ F(k+1, } ; k+ 3 ; e−25) = (2 cosh §) −(+1) F (≥k+į, įk+1; k+3; 1/cosh3 §), so that our harmonics (11) are, of course, equivalent to the harmonics (14) conventionally taken. Thus our procedure for building up solutions of Laplace's equation in this case yields, in fact, functions which are multiples of the Q functions: they have the advantage over the Q functions, however, that they more readily lend themselves to practical applications. In the same manner, if λ be negative, we have z = a cos u(1 +λ52)/5 = +d cos u{1/5√ −λ−5√ −x), p = a sin u(1—52)/5 = 3d sin u(1/5√−x+5√−λ), with d= 2a —λ, (16) and the unit -level is the oblate spheroid usually taken as 1 in the transformation. p = d sin u cosh §; 4 D z = d cos u sinh ý, น Phil. Mag. S. 7. Vol. 6. No. 40. Dec. 1928. which is the usual solution. But we may again point out that our new hypergeometric form of solution (11) allows very simple approximations to be arrived at in all the usual applications concerning oblate spheroids. It is, in fact, a serious demerit of the treatment of problems relating to spheroids by means of Q and q functions that these functions do not lend themselves to approximation and computation. We may also point out that the treatment by hypergeometric functions covers both the cases of prolate and oblate spheroids without the necessity for the separate discussion of the two cases which is called for when the Q and y functions are used. Finally, we may direct attention to the fact that the two types of coordinates introduced in (13, 16), as conventionally used in spheroidal problems, are not convenient coordinates to use if a comprehensive view of the problems is taken. We then aim at solving Laplace's equation in the form in which it is applicable to the surface of revolution given by +λn cos nu..., z = a cos u(1+λ) + λ2 cos 2u ... pa sin u(1-λ)-Ag sin 2u... -λ, sin nu........ (17) and, by developing the treatment which we have just elaborated, we are able, by a simple procedure, to construct harmonics suitable for any more general case. Indeed it appears that there is no essential difference between de veloping or correcting spherical harmonics so that they may apply to the spheroids, and developing them in a general manner so that they may apply to any of the surfaces given by the equation (17). more CXI. The Deterioration of Quartz Mercury Vapour Lamps and the Luminescence of Transparent Fused Quartz. By A. E. GILLAM and R. A. MORTON *. [Plate XXIV.] N many photochemical reactions in which mercury vapour lamps and fused quartz vessels are used, the efficiency of the processes appears to fall off with time. This may be due to the setting up of chemical equilibria, to a decrease in the output of light from the lamp, or to the development of some degree of opacity in the quartz vessels. There can be no doubt of the fact that most, if not all, makes of quartz mercury lamps deteriorate rather seriously after running for a relatively small number of hours. It is equally certain that fused quartz commonly undergoes a change under the action of light, and that this change is accompanied by some loss in transmission. Little trustworthy information is available as to the extent of these phenomena and as to the connexion which may or may not subsist between them. It is conceivable that the deterioration of the lamps is due to a change in the properties of fused quartz. The purpose of the present investigation is to study the two phenomena a little more closely. As a result of extended photochemical researches in Prof. Baly's laboratories, a large number of old lamps of the U type have accumulated, and inspection of these shows that the following are the visible signs of ageing : (1) The interior surface becomes coated with a brownish black deposit, which is especially noticeable at the thick constriction. In many old lamps the discoloration is distributed over the whole U tube, and the deposit cannot fail materially to reduce the intensity of the transmitted light. (2) In some lamps the thick quartz at the constriction appears to have been fractured internally, and globules of mercury are seen to be embedded at least a millimetre below the surface. In one case a white mass was seen to be embedded in the quartz. A number of worn out lamps were broken up and the. mercury removed as completely as possible. The black deposit proved to be strongly adherent and resisted the action of strong acids. Boiling aqua regia exerted no * Communicated by Prof. E. C. C. Baly, C.B.E., F.R.S. effect, and even after the quartz had been left in contact with aqua regia for some months the stain appeared to be unaffected. If, however, a badly discoloured fragment of quartz was strongly heated in a blow-pipe flame, the black stain was gradually replaced by a white deposit, which again resisted the action of reagents. No emission of light was seen (vide infra) during the heating process. From this it would appear that the deposit is unlikely to consist of mercury or mercury compounds, but may possibly consist of elementary silicon. It has been suggested (Drane, Brit. J. Actinotherapy, June, 1926) that in certain evacuated lamps having a tungsten or molybdenum anode, thin layers of compounds of these metals become deposited on the inner surface of the arc tubes and act as selective filters. Drane also states that " a decrease in intensity is observed as the lamp is used, due essentially to partial devitrification of the silica glass of the arc tube." Prolonged heating causes the "amorphous fused silica to change over to tridymite and cristobalite in varying amounts, depending upon the conditions of heating and the presence of catalysts, if any. In this respect the hot mercury vapour is not without influence upon the devitrification which occurs upon the inner surface of the arc tube." Drane's remarks occur in a paper on "The operation of quartz mercury vapour lamps," and are only incidental to his main theme. The detailed evidence for these views and particularly for the part played by devitrification in the deterioration process does not appear to have been published as yet. It may be of interest briefly to summarize the evidence for deterioration. The formation of ozone in the surrounding air is much more noticeable with a new lamp than with an old lamp. From this it can be inferred (cf. Lenard, Ann. Physik, 1900, i. p. 486) that the emission of very short wave ultra-violet rays decreases with time. Spectrum photographs taken with Schumannized plates show not only that the emission from an old lamp is materially less than that from a new lamp for all wave-lengths, but that the spectrum does not extend quite so far into the ultra violet. Actinometric records of the output at different stages in the history of a lamp exhibit the deterioration very clearly, and it is interesting to note that although a gradual decrease in emission is shown over the whole ultra-violet spectrum, a selective decrease is manifest in the very short wave ultra violet. The falling off is most noticeable on the hort wave side of 250 μμ. In the figure the life-history of an atmospheric burner as given by three different chemical methods of gauging ultraviolet intensity is shown (for details of these methods see Gillam and Morton, Journ. Soc. Chem. Ind. 1927, xlvi. p. 417). It will be seen that the very high initial output is maintained only for a small fraction of the effective life of the lamp, but that the decrease in intensity tends afterwards to occur much more slowly. The nitrate actinometer (ibid. p. 415) registers the greatest drop in output, a fact of some Fig. 1. 90 80 70 Decrease in output of a 230-volt atmospheric mercury vapour lamp with time. The curves represent the output as measured by: 1. Anderson and Robinson's method. 2. The acetone-methylene-blue gauge. significance, since the chemical change which is measured occurs almost exclusively with rays shorter than 270 μp. The spectroscopic and actinometric data we have obtained are perhaps a little unexpected. There are clearly two factors in the deterioration process, a shortening of the spectrum in the extreme ultra-violet and a gradual loss in 1200 |