July 12, 1888] 



NATURE 



259 



July. 

 16 



18 



20 



h. 



17 

 O 



Mars in conjunction with and 6° 40' south 



of the Moon. 

 Jupiter in conjunction with and 4 5' south 



of the Moon. 

 Mercury stationary. 



Variable Stars. 



Star. 

 U Cephei .. 



W Virginis .. 

 S Librae 

 U Coronse .. 

 W Herculis .. 

 U Ophiuchi.. 



W Sagittarii 

 Z Sagittarii.. 

 T Serpentis ., 



/3 Lyrse 



R Lyras 

 R Cygni 

 S Aquilas .. 

 S Delphini .. 

 X Cygni 



R.A. Decl. 



h. m. „ / 



o 52-4 ... 81 16 N. 



13 203 



14 55 o 



15 i3"6 



16 3i'3 



17 io - 9 



48 S. 



4 s. 



3N. 



37 34 N. 

 1 20 N. 



32 



... 17 57-9 ... 29 35 S 



... 18 14-8... 18 55 S. ... „ 

 ... 18 23-4 ... 6 14 N. ... ,. 

 ... 18 46-0 ... 33 14 N. ... ,, 



... 18 51-9 ... 43 48 N , 



... 19 33-8 ... 49 57 N. ... „ 

 ... 20 65 ... 15 17 N. ... ,, 



... 20 379 ... 16 41 N , 



... 20 39-0 ... 35 11 N. ... „ 



M signifies maximum ; m minimum. 



Meteor- Showers. 



R.A. Decl. 



h. 



July 15, 21 



,, 20, 21 



„ 16, 22 



„ 20, O 



„ 15. 21 



I, 20, 



„ 19, 2 



II 19, 22 



,, 16, 21 



,1 16, o 



II 20, 



M 15. 1 



>, 18, 



,. I9i 



,, 21, 



„ 18, 



>> 21, o 



31 m 



11 m 

 o vi 



44 m 



43 * 



m. 



4 m 



12 vi 

 o VI 

 o m 



M 

 oM 



VI 



M 



The Perseids .. 

 Near 7 Draconis 

 ,, o Lacertae 



2D 



'269 

 336 



50 N. 



51 N. 

 49 N. 



Swift, streaks. 

 Swift. 



Swift, short. 



GEOGRAPHICAL NOTES. 



The Geographical Society of Paris have decided to avail 

 themselves of the Universal Exhibition at Paris, next year, by 

 convening an International Congress of the Geographical 

 Sciences, to meet in the month of August. There will be two 

 classes of members, subscribing respectively 40 and 20 francs, 

 and each member will be entitled to receive a copy of the pub- 

 lications of the Congress and have a vote in the questions 

 discussed at the meetings. Each Society represented at the 

 Congress will be invited to submit a report on the voyages, 

 explorations, and publications which have most contributed, in 

 the country to which it belongs, to the progress of geography 

 during the past hundred years ; the combined reports will 

 afterwards be published with the names of their authors. 



Dr. H. Meyer has made some important corrections in the 

 preliminary account of his ascent of Kilimanjaro. After verify- 

 ing and correcting his barometrical observations, he admits that 

 the previously accepted height of 18, 700 feet is more accurate 

 than that given by himself, 19,850 feet. He then refers to the 

 dense mist which prevented him from seeing beyond a wall of 

 inaccessible ice, 130 feet high, which his first account indicated 

 as being the terminal point of the peak. It results from these 

 observations that Dr. Meyer did not reach to within 820 feet 

 of the summit of Kilimanjaro, which therefore still remains 

 unconquered. 



M. Jules Borelli, the French traveller, who accompanied 

 M. Rimbaud last year in his interesting journey from Antotto to 

 Harar, is engaged in exploring the country to the south-west of 

 Shoa. The Paris Geographical Society has received some of 

 the results accruing from his journey from Antotto to Jiren, 

 which is situated in 7°42' N. latitude, and 34°35' E. longitude. 

 Among these results is the discovery of the sources of the River 

 Hawash, which lie at the foot of Mount Ilfata at the extremity 

 of the Meca range, and not near Mount Dandi, as hitherto 

 supposed. On the .-ummit of the latter peak the traveller found 

 a double lake resembling in shape the figure 8, which is of con- 

 siderable extent and depth ; an affluent of the Gudar, and thus 

 of the Abbay, issues from this lake. He also discovered a deep 

 lake at the bottom of the immense crater mountain known as 

 Mount Harro ; the surroundings of this sheet of water are 

 described by the traveller as of incomparable beauty. From 

 this lake, which is named by the natives Wancit, a stream issues 

 and joins the Walga, the source of the latter river being in the sum- 

 mit of Mount Harro. Dr. Traversi, the Italian explorer, made in 



June, 1887, an excursion into the mountainous region of Urbanagh,. 

 lying to the east of the district now being explored by M. 

 Borelli. The chief result of this journey of Dr. Traversi is to 

 throw light on the problem of the hydrographical systems of the 

 Somali and Galla countries. From the summit of Mount 

 Gafat he was able to comfirm his previous observations made 

 near the Suai Lake, with reference to the three lakes above- 

 mentioned and their interconnection. 



ON CERTAIN INEQUALITIES RELATING TO 

 PRIME NUMBERS. 



T SHALL begin with a method of proving that the number of 

 L prime numbers is infinite which is not new, but which it is 

 worth while to recall as an introduction to a similar method, by 

 series, which will subsequently be employed in order to prove 

 that the number of primes of the form 4« + 3, as also of the 

 form 611 + 5, is infinite. 



It is obvious that the reciprocal of the product 



•'•I' 



A/V pj\ pj \ Pv.p, 



(where/, means the uh in the natural succession of primes, and 

 p s means the highest prime number not exceeding N) 1 will be 

 equal to 



i+h + h+\ + l + 1s+ . . . . +n+ R > 



and therefore greater than log N (R consisting exclusively of 

 positive terms). 



Hence 



1 + — 

 A 



1 + - 

 A 



1 + - ) > M log N, 



An./ 



where 



M = 1 - 



'Nj» 



and is therefore greater than -. 



IT 



Hence the number of terms in the product must increase 

 indefinitely with N. 



By taking the logarithms of both sides we obtain the 

 inequality 



Si - |S 2 + iS 3 - iS 4 + . . . . >loglogN, 



where in general S/ means the sum of inverse s'th powers of all 

 the primes not exceeding N ; and accordingly is finite, except 

 when i — \, for any value of N. We have therefore 



Si -> 1°§ l°g N + Const. 

 The actual value of Sj is observed to differ only by a limited 

 quantity from the second logarithm of N, but I am not aware 

 whether this has ever been strictly proved. 

 Legendre has found that for large values of N 



(I -.«■(! 



Consequently 



(■-;,)(' 



This would show that the value of our R bears a finite ratio 

 to log N ; calling it 6 log N we obtain, according to Legendre's 

 formula, 



which gives 6 = *8n, 



_ 1) ( l _ - 1 \ = I ' I °4 



' ' 'V Pn. p ) log N' 



~ PJ ' ' ' \ ~ Ps.J 



•552 



log N* 



I + 



= -552, 



so that the nebulous matter, so to say, in the expansion of the 

 reciprocal of the product of the differences between unity and 

 the reciprocals of all the primes not exceeding a given number, 

 stands in the relation of about 4 to 5 to the condensed portion 

 consisting of the reciprocals of the natural numbers. 



I will now proceed to establish similar inequalities relating to 

 prime numbers of the respective forms 4« + 3 and 6;/ + 5. 



Beginning with the case 4« + 3, I shall use qj to signify the 

 yth in the natural succession of primes of the form 4« + 3, and 

 </ N to signify the highest q not exceeding N, N.^ itself 

 signifying the number of q% not exceeding N. 



1 N/ itself of course denotes in the above notation the number of primes- 

 (/) iut exceeding N. 



