526 



NATURE 



[March 30, 1893 



SOCIETIES AND ACADEMIES. 

 London. 

 Mathematical Society, March 9.— Mr. A. B. Basset, 

 F.R.S., Vice-President, in the chair.— Mr. T. J, Dewar ex- 

 hibited, with the aid of a stereoscope, twenty stereographs of the 

 regular solids. These were not photographs of a solid object 

 from two points of view for binocular vision, but the same object 

 was drawn twice over by Mr. Dewar in perspective with different 

 station points. The relief was aided by making the lines in the 

 foreground thick, and those behind thin.— Mr. Love read a note 

 on the stability of a thin rod loaded vertically. Suppose a thm 

 rod or column is held vertically at its lower extremity, and 

 loaded at its upper extremity. It is well known that, unless the 

 load exceeds a certain limit, the rod will be simply compressed 

 longitudinally without being bent. If, however, the limit is ex- 

 ceeded there exists a curved form in which the rod can be held 

 by the application of the given load. This form must belong to 

 the elaslica family of curves. Now when the length and the load 

 are given the elastica is not entirely determinate. In fact for the 

 same length and the same load (if sufficiently great) there exist 

 forms having respectively i, 2, 3. . . . inflexions. These are 

 the curves figured in Thomson and Tail's "Nat. Phil.," part 11. 

 p. 148, and for our present application the rod must be supposed 

 held at the middle point of one of the bays, into which it is 

 divided by the line of action of the load. Thus the part of the 

 curve between the point of support and the nearest inflexion is 

 half a bay, the rest of the curve up to the point of attachment of 

 the load consists of an integral number of complete bays. Now 

 although all these forms are possible there is only one which is 

 stable, and that is the form with a single inflexion. To prove 

 this we have to investigate the potential energy in the configura- 

 tion with a single inflexion, in which the curve forms a single 

 half bay, and in the configuration with 211 + i inflexions, in which 

 the curve forms « + i bays. It is not difficult to prove that in 

 every case the latter potential energy is the greater. It follows 

 that the figures given by Euler's " Theory of Struts " in which 

 the rod forms a curve w hich is nearly a curve of sines of small 

 amplitude crossing the line of action of the load more than once 

 are all unstable iorms. The stable form is a curve of finite 

 •curvature, which never crosses the line of action of the load. 

 — Prof. Lloyd Tanner next made a communication on complex 

 primes formed with the fifth roots of unity. The object of the 

 paper is to explain a method of calculating the complex prime 

 factors of real primes included in the form lO/t-fi. The only 

 published method which I have met with is due to Kummer. 

 This is not restricted to the particular case here considered ; but 

 as it involves the determination of the G.C.M. of two complex 

 numbers, it is probably more laborious than the method now 

 communicated. The method adopted by Reuschle in the cal- 

 culation of his tables does not appear to have been published. 

 The process here is based on the indeterminate equation 



X2-5Y2 = 4/. 



A minimum solution of this equation gives the "simplest 

 prime factor according to Rummer's definition {Berlin Monats- 

 berichte, 1870, p. 413) and solutions in which Y is a multiple of 

 5 give the "primary" prime factors which Kummer found it 

 necessary to use in order to establish the general law of reciprocity. 

 In solving the equation Lagrange's method turns out to be 

 impracticable, and a short discussion— treated graphically — is 

 introduced, which is sufficient to show the relations between the 

 different solutions. These relations can be expressed in the 

 form — 



(:::)('^'^)=(::f)^^') 



and it is interesting to note the intimate connection between 

 these matrices and the complex units. From any solution 

 <X, Y) three numbers Ao, A^, Aj, are found, Ao being the integer 

 next greater than 2X/5, and these serve to determine the values 

 and sequence of the co-ordinates a^,, a^, &c., in the required 

 prime factor 



The first condition is 



Ao = flo^ + «i2 + a^' 

 NO. 1222, VOL. 47] 



+ a{- -i- ^4-. 



The values of a have to satisfy other conditions, some of which 

 are tested by mere inspection. To give some idea of the facility 

 of the method from the calculator's point of view it may be 

 stated that the determination of the prime factors of two primes 

 selected at random in the second million (viz. 1,562,051 and 

 1,671,781) was completed in three hours. The only auxiliary 

 table required is a table of squares : and if this extends to the 

 square of 7000 it will suffice for the factorisation of all primes in 

 the first nine millions. Tables are appended giving the simplest 

 —and simplest primary— prime factors of all suitable primes less 

 than 10,000 The reciprocal factors are also given after the 

 first thousand. For the first thousand the reciprocal factors have 

 already been published ; and instead of giving these again, a 

 comparison is indicated between the factors here given and those 

 published in Reuschle's tables. The result of the comparison 

 suggests that Reuschle's method of calculation was not the same 

 as that now communicated.— The dioptrics of gratings, by Dr. J. 

 Larmor.F.R.S. Whenabeamof light falls on a continuously ruled 

 or striated surface, in addition to the principal portion that passes 

 on and the portion that is scattered and lost by the roughiiesses of 

 the surface, there are formed a series of secondary diffracted 

 beams that are propagated onward in oblique directions. Each 

 of these beams is produced in the well-known manner by the 

 union of the elements from the different striations (or homo- 

 logous groups of striations), which arrive at its front in a common 

 phase. The dioptrical discussion of such diffracted beams, that 

 is so far as regards their 'geometrical properties, forms a rather 

 simple case of the theory of the refraction of a general dioptrical 

 pencil, which has been developed by Hamilton, Maxwell, and 

 other writers. In the case of homogeneous wave-length A, when 

 the principal beam, coming from its focal lines, is refracted at 

 the striated surface to two other focal lines, the «th diffracted 

 beam is propagated as if it were simply refracted at a new 

 surface formed by adding on at each point a thickness (yii - i)«otA 

 of the refracting medium in front of the original surface ; where 

 m is the number of striations counting from any aibitrary origin 

 on the surface up to the point. The case of reflexion is included 

 by making ju, = — i. Asa special example, it is well known that 

 the positions of the primary and secondary foci for conical pencils 

 in a spherical Rowland grating, are determined by the same 

 formulae as hold for reflexion in a curved mirror. The treat- 

 ment of the aberration at the focal lines, or the discussion of the 

 caustic surfaces of the diffracted beams, is reduced immediately 

 to the Hamilionian formulae by noting that the characteristic 

 function of the beam is increased by the quantity (m - \)ntn\, 

 exactly, in crossing the diffracting surface. — The secretary read 

 a brief abstract of a note by Prof L. J. Rogers on a three-fold 

 symmetry in the elements of Heine's series. — Messrs. Greenhill, 

 Walker, Cunningham, and the Chairman joined in the dis- 

 cussions on the papers. 



Royal Microscopical Society, March 15. — A. D. 

 Michael, President, in the chair.— The president said that a 

 series of thirty-six photomicrographs had been sent to the 

 Society of Arts, in compliance with the request read at the last 

 meeting, for exhibition at Chicago.— An electric turntable was 

 exhibited on behalf of Mr. Payne, of Newcastle. It consisted 

 of a brass turntable of ordinary pattern having an electric motor 

 fitted beneath the plate ; the whole was caused to revolve by the 

 current from a bichromate battery cell,— Dr. W. H. Dallinger 

 gave a brief description of Prof. Biitschli's experiments on the 

 so-called artificial protoplasm ; and said in conclusion, that he 

 could not suppose that any one looking at these forms would 

 regard them as in any way allied to living matter. The more 

 intimately they became acquainted with them the more 

 sure they would become that' they were only forms, and that 

 those which appeared under a low power to be so much like 

 tissue were under a high power seen to be minute bubbles and 

 nothing more. He believed the movements observed would be 

 found to be due to the effect of differences of surface tension, 

 and that the study of them would no doubt help them to under- 

 stand some of the mechanical properties of protoplasm, but they 

 did not leave an impression that" they had caused an approxi- 

 mation in the least degree towards the artificial production of 

 protoplasm.— Mr. R. T. Lewis exhibited and described a new 

 species of Aleurodes [A. asfaragi) which had been found upon 

 the leaves of asparagus in Natal.— Mr. T. F. Smith read a note 

 on the use of monochromatic yellow light in photomicrography. 

 —Prof. F. Jeff"rey Bell read a note from Dr. A. M. Edwards on 



