76 



NATURE 



\Nov. 23, 1871 



primes "among Its factors, then the corresponding factors 

 are to be omitted out of the product. We thus see that 

 if two even numbers of considerable magnitude lie adjacent 

 or tolerably near to each other, one of which is the double of a 

 prime, but the other six times a prime, the number of preter- 

 prinics relative to the latter will be about twice as many as 

 those relative to the former. For the pui-pose of greater sim- 

 plicity of expUnation, the formula of approximation has been 

 stated above with less accuracy than it admits of being stated 

 with. Instead of the total number of odd primes being multi- 

 plied by the product of factors last described, those only should 

 have been taken which are not intermediate between 2 and sjn, 

 and the result so modified should have been stated to be the 

 probable value not of the total number of preler-primes, but 

 only of such of them (by far the larger number) as are not of 

 the excluded class above described, nor subtracted from «, give 

 rise to remainders belonging to such class. The author has found 

 by actual trial on an extensive scale, that the estimated values of 

 the number of decompositions never differ by more than a 

 moderate, and in some cases exceedingly slight, percentage from 

 their actual values determined by the use of Borchardt's tables. 

 The same methods enable him also to assign a probable value to 

 the number of modes of resolving an odd number into the sum 

 of o le prime and the double of another, and in general lead to 

 an approximate representation of the number of solutions in 

 prime numbers of any system of linear equations of which the 

 total number of solutions is limited, and even to resolve approxi- 

 mately such questions as that of determining how many prime 

 numbers there are inferior to a given limit, which are followed by 

 prime numbers differing from them by any assigned interval. 

 Since the communication made to tlie Mathematical .Society, the 

 secretaries have been favoured with a note from wlvch they un- 

 derstand that Dr. Sylvester has verified his results by quite a 

 different method. The exact numlier of the solutions of the 

 equation x ^ v = n in prime numbers may be expressed alge- 

 braically by means of the method of generating functions in terms 

 of the inferior primes to n. The expression will be found to 

 consist of two parts, one a constant multiple of it. the other, a 

 function of the roots of unity corresponding to the several inferior 

 primes and their combinations. The former non-periodic part 

 may obviously be regarded as the even value of the expression, 

 and Dr. Sylvester has found that it is identical with the value 

 obtained by the method of averages previously employed. In 

 order to prove strictly Euler's theorem, it only remains to show 

 that the entire expression can never become zero. This Dr. 

 Sylvester believes he has the means of doing, and at the same 

 time of a.ssigning exact limits to the number of soUiiions in 

 question ; but in a matter of so much moment, and of such singu- 

 lar interest, does not wish to express himself in a more decided 

 manner, until he has had the opportunity of subjecting his 

 method to a further rigorous examination. 



Royal Astronomical Society, November 17. — Mr. W. 

 Lassell, president, in the chair. The Astronomer Royal showed 

 a drawing of Encke's comet made by Mr. Carpenter of Green- 

 wich ; it gave the inipres-sion of a somewhat shuttlecock-shaped 

 nebulous haze, with two wings of much fainter light, extending 

 on either side, giving a flattened appearance to the head of the 

 comet. Dr. Muggins made a drawing which coincided in all 

 e.-sential particulars with that of Mr. Carpenter. He thought 

 that he had detected a very minute but distinctly-marked nucleus 

 in the paraboidal-shaped head of the shuttlecock. The whole 

 light of the comet was very faint, but he had succeeded in 

 ob aining its spectrum, which, as in the case of that of Comet II, 

 1868, consisted of three bands, apparently identical with the 

 bands in the soe trum of the vapour of carbon. The middle band 

 situated near " little b " was much brighter than than the o her 

 two, and he was quite satisfied of its identity with the middle 

 bands of carbon vapour ; the two outlying bands were much too 

 faint for him to speak with confidence of their identity, but they 

 appeared to correspond. The Astronomer Royal showed a 

 celestial globe, on which he had fixed a small white wafer in the 

 place occupied by the sun, and a piece of white paper cut out 

 to represent the comet. He pointed out that its longer axis was 

 directed almost exactly to the sun, and that its head and nucleus 

 were turned away Irom the sun. This appears to be the almost 

 universal rule with the smaller class of comets. Unlike the sheep 

 of Httle Bo Peep they carry their tails before them, and not until 

 their smaller fan-shaped ap|jendages have been well warmed try the 

 sun's rays, do they begin to shoot out large tails in the other 

 direction. — A paper was read by Prof. Grant, in which he 



pointed out that as eariy as the year 1S52 he had recognised the 

 continuity of a red envelope enclosing the sun, of which the 

 prominences were merely the more elevated portions ; he had 

 come to this conclusion from a comparison of the observations 

 made during the total eclipses of 1842 and 1851.— A discussion 

 then followed as to whether there were any permanent markings 

 upon Venus. Dr. W. De la Rue and Mr. Browning affirmed 

 that they often saw spots and other irregularities of surface. The 

 authority of Mr. Dawes, and many other observers of note, was 

 cited to the contrary. — Some careful drawings of the Zodiacal light 

 as seen by Captain Tupman while cruising in the Mediterranean 

 were handed round. It was pointed out by Mr. Ranyard that 

 the axis of symmetry of the light was in many instances greatly 

 inclined to the ecliptic, and that the distance of the node of the 

 axis from the sun was in some instances more than 40'. 



BOOKS RECEIVED 



English —The Geology- of Oxford and tlie Thames Valley : J. Phillips 

 (Macmillanand Co.)-— Weale's Treatises; Rudimentary Geo'ogy ; Histori- 

 cal: R. Tale (l.ockwood and Co )— Profitable and Ornamental Poultry: 

 H Piper (Gro-imbridEe and Sons) — Ganot's Elementary freali eon Physics, 

 Exp;rimental and Applied: TranslUed by G Atkinson. 5th edition (Long- 

 mans and Co.).— Tables of Velocity, Time of Flight, and Energy of Various 

 Projectiles ; Bashlbrth Chronograph (E. and F. bpoa) —The Discovery of a 

 New World : G. Thomson (Longmans and Co ), 



FriREiGN —(Through Williams and Norgate )— Les Migrations Humaincs 

 en Oceanic d'apres les fails naturelles : Jules Gamier. 



THURSDAY, November 23. 

 RoVAL SociETV, at 8.30. — On the Behaviour of Supersaturated Saline 



Solutions when Exposed to the Open Air: C Tomlinson, F.R.S.— On 



Experimental Determination of the Velocity of Sound : E. J Stone, 



F.K.S ; (i) Second Paper on the Numerical Value of Euler's Constant. 



&c. ; (2) Second Paper on the Numerical Values of e, log e^, log e^, log e^, 



and log ('", &c. ; W. Shanks. 

 SociETV OF Antiquaries, at 8.30. — On Medi<deval Representations of the 



Months and Seasons: James Fowler, F.S.A. -On some Casts of Ivories 



from Cologne: Augustus W. Franks 

 London Institution, at 7.30. — The Influence of Geological Phenomena on 



the Social Life of the People : Harry G. Sceley, F G.S. 



FRIDAY, November 24. 

 QuEKETT MiCROSCOflCAL Club, at 8. — On the Minute Structure of Tre- 

 melloid Uredines : JM. C. Cooke. 



MONDAY, November 27. 



RovAL GEOGRAPUlrAL SociETV, at 8.30.— Exp'oration of the Volcanic 



Districts East of Damascus : Capt. R. F. Burton.— Journey in Southern 



Arabia : Baron de Maetzan. 



London Institution, at 4. — Smell, Taste, and Touch : Prof. Huxley, 



LL.D., F.R.S. (Course on Elementary Physiology). 



WEDNESDAY, November 29. 



Society of Arts, at 8 -On Tramways and their Structure, Vehicles, 



Haulage, and Uses : W. Bridges .\dams. 

 Arch.-iiologic.'VL Association, at 8. 



THURSDAY, November 30. 

 RoVAL Societv, at 8.30. — President's Address. 

 SociETV OF Antiquaries, at 8.30. 



London Institution, at 7.30. — Science and Commerce, illustrated by the 

 Raw Materials of our Manufactures. (II.) P. L. Simmonds, F. R.C^.I. 



CONTENTS 



: FOR Wome 

 Ma 



Our Book Shelf 59 



Letters to the Editor : — 



Oceanic Circulation.-Dr. William B. Cari'ENTEr, F.R.S. . . 60 



The Solar Parallax —Prof. Simon Newcomb 61 



The Aurora of Nov. 9 and ic— Rev. H. C. Key; Alfred W. 



Bennett, F.LS 61 



TheGhostofFlamstead.—R. A. Proctor, F.R.A S 61 



Creators of .Science. - Dr C. M. Inglebv 62 



Descartes' "Animated Machines." — Rev. J. P. Mahaffv .... 62 



Plmc-Direction.— Prof. J. D. Everett 63 



" Wormell's Mechanics."— R. Wormell 63 



One o' the Greatest Difficulties of the Darwinian Theory. 



By Dr. Lionel S. Bealh, F.R.S 63 



On the Recurrence of Glacial Phenomena during Great 



Continental Epochs. By Prof. A. C. Ram.sav, F.R.S 64 



Wood's " Insects at Home." {IVith Illustrations.) 6; 



; Laws of Currents in ORDtNARY Condut 



Not 

 Cold 



the Sea 1 7, 



Physiology for Women. By Prof. Bennett 73 



Societies and Academies 74 



Books Received 76 



Diary 76 



