58 



NATURE 



[March io, 1910 



provide maintenance. Many localities feel they have done 

 all they can, and they also feel they are not merely doing 

 local work, but national and Imperial, indeed, world-wide 

 work. Students are drawn from every part of the Empire 

 and from foreign countries, particularly China and Japan, 

 and they are under no obligation to give their services 

 where they are trained. Any increased grant now given 

 by the nation will be used, not in the fixed and ordinary 

 work of the institutions, but in the highest class of work 

 and in various enterprises that are being kept back for 

 want of funds. The speakers also pointed out that there 

 is under present conditions a certain amount of wasteful- 

 ness, not in money, but in brains and energy, because at 

 their meetings the authorities are generally occupied, not 

 in discussing how best to spend the money and what under- 

 takings will be best for the country, but merely how to 

 economise their funds and how to save loL or 50Z. Work 

 is lying ready at hand which they are powerless to under- 

 take. Mr. Lloyd George, in the course of a sympathetic 

 reply, told the deputation they could not have come at a 

 worse time. Nothing definite was settled, but a committee 

 has been appointed by the deputation to prepare more 

 detailed information for the Chancellor of the Exchequer 

 as to the financial requirements of the various institutions. 



SOCIETIES AND ACADEMIES. 

 London. 

 Royal Society, January 27. — Sir James Dewar : Long- 

 period determination of the rate of production of 

 helium from radium. In a previous communication the 

 rate of the production of helium from 70 milligrams of 

 radium chloride was determined by a succession of observa- 

 tions on the growth of pressure measured by a McLeod 

 gauge. These observations extended over a period of about 

 six weeks. It was thought desirable to make an experi- 

 ment to determine the amount of helium resulting from 

 this same sample of radium, after standing in a sealed 

 bulb for an extended period. For this purpose the bulb 

 containing the radium chloride was sealed off at the con- 

 clusion of the above-mentioned experiment of 1908 and 

 kept for nine months. In order to measure the helium 

 thus produced it was necessary to devise a vacuum-tight 

 joint between the sealed radium bulb and a McLeod gauge 

 so constructed that, after thoroughly exhausting the gauge, 

 the drawn-out end of the radium bulb could be broken off, 

 thus allowing the pressure of the accumulated helium in 

 the radium bulb to be rapidly determined. The total 

 volume of the apparatus was 320 c.c. The pressure in the 

 radium bulb when sealed off at the conclusion of first ex- 

 periment was 0-00406 mm., the partial pressure due to 

 this amount of helium would be 000008 mm., which must 

 be deducted from the observed pressure to get the true 

 pressure due to the helium produced in the radium bulb 

 during the period in which it remained sealed up ; also the 

 pressure in the gauge, before breaking (0-00005 mm.), must 

 also be deducted. This gives a corrected pressure of 

 0-01613 mm., obtained after heating the salt, due to the 

 helium produced from 70 milligrams of pure radium 

 chloride during a period of 275 days, in a space the total 

 volume of which was 320 c.c. The value of the rate in 

 terms of cubic millimetres of helium per gram of radium 

 per day is thus deduced as 0-463. 



March 3. — Sir Archibald Geikie, K.C.B., president, in 

 the chair. — T. G. Bedford : The depression of freezing 

 point in very dilute aqueous solutions. — J. Mercer : Sturm- 

 Liouville series of normal functions in the theory of 

 integral equations. It is the purpose of this memoir to 

 develop the theory of Sturm-Liouville series of normal 

 functions as a branch of the theory of integral equations. 

 In the first place, two theorems are established relative to 

 the series 



positive type in the square Q defined by a<s<b, a<t<b : the 

 normal functions are assumed to have such an order that the 

 singular value corresponding to yfi„(s) does not decrease as n 

 increases : no restriction is placed upon j{s) beyond that^t 

 should have a Lebesgue integral in {a, d). Denoting by Kx{s, i) 

 the solving function corresponding to K(j, t), the first theorem 

 is to the effect that the upper and lower limits of indeterminacy 

 of the above series include 



TrnT" - x/^^^^(^' ')/<')'^' 



between them. 



According to the second 



+ <^: 



n(s) f''^n{i)f{ 



(t)dt + 



in which ^-^(s), ^<i{s), . . ., »^i,(j), . . . are a complete system 

 of normal functions corresponding to a function (K(j, /)) of 



NO. 2106, VOL. 83] 



lim - X f Ka(j, t)f{s)f{t)(dsdt) 

 exists and is equal to the sum of the series 



. . . +[[W)/"(/K/]% . . ., 



when the latter is convergent ; whilst the limit is + 00 , when 

 the series is divergent. It is then shown that, when K(j, i) is 

 the Green's function of 



satisfying a pair of boundary conditions at the end points of 

 (o, It), an asymptotic formula for YL\(s, t) exists which permits 

 the deduction of important theorems relative to the canonical 

 Sturm-Liouville series 



. . . +U^) [''U^)f{()dt+ ... 

 J 

 The normal functions ^i(s), <f'.,(i), . . ., ^n{s), ... are now 

 solutions of 



ds^ 



which, for suitable values of A, satisfy the same pair of 

 boundary conditions as K(s, t) ; to particular systems _ of 

 these functions correspond Fourier's sine and cosine series. 

 The results obtained for any canonical Sturm-Liouville 

 series are very similar to, but slightly more general than, 

 those for the two particular series which are associated 

 with the names of Fej^r, Hurwitz, and Lebesgue. The 

 fourth section of the memoir is devoted to an investigation 

 of the convergence of canonical Sturm-Liouville series. In 

 the course of this, it is shown that the convergence of any 

 one of these series at a point of the open interval (o, ir) 

 involves the convergence of all the other series which corre- 

 spond to the same function fis). The memoir contains an 

 extension of all results obtained for the canonical to the 

 most general type of Sturm-Liouville series. — A. Von 

 Antropoff : The solubility of xenon, krypton, argon, neon, 

 and helium in water. — L. N. G. Filon : Measurements of 

 the absolute indices of refraction in strained glass. If 

 light be transmitted through a slab of glass under tension 

 T in a direction perpendicular to the line of stress, it is 

 broken up into two components, polarised in planes per- 

 pendicular and parallel to the line of stress. If fi be the 

 index of refraction of the glass in the unstrained state, 

 then, in the strained state, the indices of refraction corre- 

 sponding to the above two components are fi + C^T, n + C^T 

 respectively. The coefficients Cj, C^ are spoken of as the 

 stress-optical coefficients for the two rays. The present 

 paper gives an account of measurements of Cj and Cj 

 according to a method described by the author in Roy. 

 Soc. Proc, A, vol. Ixxix., pp. 440-2. The measurements 

 have been carried out on two Jena glasses bearing cata- 

 logue Nos. O. 935 and VV. 3199 respectively, the first 

 being a borosilicate, the second an " ultra-violet " glass. 

 So far as is known, this is the first series of absolute 

 measurements of Cj and C, extending fairly continuously 

 throughout the spectrum. The only previous measurements 

 are due to Pockel C4nn. d. Phys., 1902), and give the 



