6i8 



NA TV RE 



[April 26, 1894 



creasing the amplilude of the harmonic motion. Not only was 

 the machine useful for Fourier expansions, but by giving suitable 

 motions to the tangent plane developments of arbitrary functions 

 in spherical harmonics, Bessel's functions, Lame's functions, and 

 other normal forms could be determined. He had designed a 

 machine which, on Prof, Henrici's principle, develops arbitrary 

 I'unctions in Bessel's, and hoped to have shown it in working 

 order at the meeting ; the Easter holidays had prevented its 

 being finished in time. In this machine the motion is given to 

 the table by a cam and roller, the cam being shaped so that 

 the displacement of the table is .r x J(x) when the shaft turns 

 through an angle proportional to ,\. The revolving cylinder is 

 driven by varialile gearing from the cam shaft. By using cams 

 of othfvr shapes, developments in many normal forms may be 

 obtained ; the machine is therefore of general analytical use. 

 An example of development in Bessel's worked out arithmetically 

 by two of his students, Messrs. Hunt and Fennel, was given, 

 and the process of performing the integration by the machine 

 described. Prof. Boys, speaking of arithmometers, said Prof. 

 Selling's machine had several inconveniences. In the first place, 

 it occupied a large space, and the projecting racks were apt to 

 upset things put bzhind the machine. Secondly, the result of 

 any operation was indicated by continuous motion, and therefore 

 cannot be readoff instantly with certainty. On the other hand the 

 "Brunsviga" machine vvasvery compact and convenient, the only 

 serious dcftct being that one cannot carry on figures obtained as 

 the result of one operation to work with again, as was possible 

 in the well-known Colmar machine. As another improvement 

 he suggested that the two sets of numbers on the wheels show- 

 ing the result of any operation, should be coloured differently, 

 so that it would be easy to see whether multiplication or division 

 had been performed. The labour of operating with large digits 

 could then be considerably reduced with certainty. For example, 

 in muliiplying by 2998, instead of 28 (2 + 9 + 9 + 8) turns of the ! 

 handle, 5 would be sufficient, viz. 3 in the forward direction 1 

 and 2 backward, thus giving 3002. In his opinion lo^arithum 

 tables were not nearly so convenient for ordinary calculations 

 as this machine. — Mr A. P. Trotter described how, by the use j 

 of templates cut to suitable shapes, one could obtain true curves ! 

 from those given by recording voltmeters and similar apparatus, j 

 Mr. Yule said he had recently seen the newest analyser made i 

 by Coradi for Prof Weber, and was present when it was tested 1 

 by the latter on a simple harmonic curve. It gave excellent 

 results, the errors not amounting to i part in 2000. Speaking of j 

 the " hatchet " planimeter, he thought the first one was exhibited j 

 by Mr. Goodman at the Institution of Civil Engineers. Mr. A. I 

 Sharp, he said, remarked that since last meeting he had designed | 

 an inversion of the mechanism in his harmonic analyser, which 

 made it much more practical. Prof. Henrici, in reply, said the 

 uses of his first machine, suggested by Prof. Perry, might lead 

 to great developments in this subject. Lord Kelvin had shown 

 that with the sphere and roller integrator products of two func- 

 tions such as /(jr) Y{x) dx could be got. Referring to Pr')f. Boys' 

 criticism on the Selling arithmometer, he did not consider the 

 difficulty in reading off the result at all serious. Mr. Trotter's 

 method of solving problems by templates might be very useful. 

 Speaking of the "hatchet" planimeter, he said he believed it 

 was first brought out in Denmark. Mr. F. W. Hill, of the 

 City of London School, had sent him a solution of its action. 

 Mr. Sharp, he said, had made a very considerable improvement in 

 his machine, and the elements of this integrator may be useful for 

 other purposes. — Mr. P L. Gray read a paper on the minimum 

 temperature of visibility, describing experiments made to find 

 the lowest temperature at which bright or blackened platinum 

 becomes visible in the dark The instrument used was a Wilson 

 and Gray's modification of Joly's meldometer, in which a thin 

 strip of platinum, about 10 cm. long and i cm. wide, is heated 

 by an electric current. The expansion of the strip is indicated 

 by an optical method, and used for estimating the temperature 

 of the strip. To calibrate the arrangement, small particles of 

 substances having known melting points were placed on the 

 strip, and observed through a microscope, the position of the 

 spot of light showing the expansion being noted when the sub- 

 stance melted. The general conclusions arrived at are : — (i) 

 That the minimum temperature of visibility is the same for a 

 bright polished surface as for one covered with lampblack, 

 although the intensity of radiation in the two cases may be dif- 

 ferent. (2) That the visible limit at the red end of the spectrum 

 varies greatly for a normal eye according to its state of prepara- 

 tion. Exposure to bright light diminishes the sensitiveness of 



NO. 1278, VOL. 49] 



the eye, and darkness increases it. (3) That for the less sensi- 

 tive condition, the minimum temperature of visibility for the 

 surface of a solid is about 470° C, but this may be much re- 

 duced by even a few minutes in a dark room. (4) That at night 

 a surface at 410° C. is visible, and that by resting the eyes in 

 complete darkness this may be reduced to 370° nearly. (5) 

 That different people's eyes differ somewhat in their " minimum 

 temperature of visibility," but probably not to any great extent 

 if tested under the same conditions as to preparation, &c. To 

 most observers the strip at these low temperatures had no ap- 

 pearance of red, but looked like a whitish mist. Inserting a 

 plate of glass or a layer of water in the line of vision had no 

 effect on the temperature of visibility. Mr. Blakesley inquired 

 if the author had tried condensing the light from the strip? As 

 to colourlessness, he observed that the parts of the retina active 

 in oblique vision were less sensitive to colour than the central 

 portions. Dr. Burton remarked that in the experiments, the 

 presence of light and not colour was being observed. When 

 illumination wa- faint, as in twilight or mocjnlight, it was very 

 difficult to distinguish colours. In the solar spectrum one did 

 not see any whitish termination at the red end. Mr. Elder said 

 Captain Abney had shown that all colours appear grey when of 

 small intensity. The President thought the question as to 

 whether visibility depends on wave-length or on energy was an 

 important one. Probably a minimum amount of energy was 

 essential. At such low temperature the emission curves of the 

 different wave-lengths may not have become sufficiently separ- 

 ated to be distinguished. Mr. Gray, in reply, said Prof. Langley 

 had shown that a minimum, but very small, amount of energy 

 was necessary to vision in all parts of the spectrum. — Dr. Bur- 

 ton's paper on the mechanism of electrical conduction was 

 postponed. 



Mathematical Society, April 12. — A. B. Kempe, F.R.S., 

 President, in the chair. — The following communications were 

 made: — On regular difference terms, by the President. (Prof. 

 Greenhill, F. R.S., Vice-President, pro tern., in the chair.) 

 In the expression of the invariants of a binary quantic Q« in 

 terms of the roots, we employ functions such as 



2(T), 



where T is a product of differences of the roots into which each 

 root enters the same number of times, and the summation ex- 

 tends to all expressions derivable from T by transpositions of 

 the roots. If the number of roots be n, and each root enters v 

 times into T, then T is a regular difference ter)ii of the system 

 of roots considered, and is said to be of degree n and order v. 

 For a given degree n the simplest regular difference terms are 

 of order i or 2, according as « is even or odd, and are called 

 elemental terms of the system of roots. The object of the paper 

 is to i-how that every regular difference term of a given system 

 of roots is a rational integral function of the elemental terms of 

 that system. One result of this theorem is that every invariant 

 of the binary quantic Q,,, which is a rational integral function 

 of the roots of Q,„ is expressible as a rational integral function 

 of such of those invariants as are of the form 



2(Ei%'^E3'' ..) 



where Ej, Eg, E3, ... are elemental terms of the n roots of Q«. — 

 Theorems concerning spheres, by S. Roberts, F. R.S. — Second 

 memoir on the expansion of certain infinite products, by Prof. 

 L. J. Rogers. — A property of the circum-circle, ii., by Mr. R. 

 Tucker.— A proof of Wilson's theorem, by Mr. J. Perott. — On 

 the sexiic resolvent of a sextic equation, by Prof. W. Burnside, 

 F. R. S. The group of an irreducible equation of the fifth degree, 

 after adjunction of the square root of the disciiminant, is either 

 the ico^ahedral group, the dihedral group for « = 5, or the 

 cyclical group for « = 5 ; the two latter being sub-groups of the 

 former. In the two latter cases the equa;ion is solvable by 

 radicals, and in the former not. For a given equation with 

 numerical coefficients the two latter cases may be distinguished 

 from the former by constructing the sexiic resolvent and deter- 

 mining whether or no this has a rational root. This sextic re- 

 solvent has been calculated by Cay ley ("Collected Papers, 

 vol. iii. 2) for the general quintic. When the quiniic is taken 

 in its standard form, x^ + ux + v = o, the calculation is 

 enormously sim(>lified (see C. Runge, Acta Math. vol. vii.). 

 For a given irreducible sextic there i^ a greater ram^e of possi- 

 bilities After adjoining the square root of the discriininant, 

 the group ol the equation may be ei her the alternating group 

 of 6 variables, a transitive group of 6 variables which is iso- 



