No'i'EMBEB 2, 1891.] 



KNOWLEDGE 



2 LI 



we are told a certain number of things can be arranged in 

 more than a decillion of ways, or, in mathematical lan- 

 guage, that the factorial of a certain number is upwards 

 of a" decillion, our curiosity is equally satisfied m being 

 told that the number would take more than (;0 figures to 

 write it down ; which, again, is the same thing as saying 

 that its logarithm is greater than 60. Such being so, 

 all we really require in these cases is a simple formula, by 

 means of which we can obtain the integral part of the 

 logarithm — the error being only in the fractional part. 



Such a formula I think I have discovered, but whether 

 already known I cannot say. It may briefly be stated 

 thus :— That the factorial of any number n is approxi- 

 mately, in the above manner, equal to 



(•37 » + l)" 

 This formula may be used, say from IS to (i.oO. 



To show that this is really the case, I give below the 

 logarithms of a few factorials, together with those of 

 the corresponding values by the formula. 



Niuuber. Loij. of Factorial. Lo,t;. of Foriuu 



13 9-79 9-9:5 



10 1.3-32 13-i2 



21 19-71 19-80 



40 47-91 47-95 



81 120-7(i 120-77 



100 l.")7-97 ]o7VX 



(il4-49 ()l4-77 



140810 14<J8-98 



Dill". 

 014 

 0-10 

 009 

 004 

 001 

 0-01 



300 

 600 



0-88 



When the factorials of members (ireatcr than 6.50 are 

 required, it is better to substitute -368 for -37 in the 

 formula. This will give a more extended range ; thus — 



Between 15,000 and 20,000 the error commences to 

 extend into the characteristic or integral part of the 

 logarithm. Accordingly, if the same nearness is required 

 for still greater numbers, another fraction would have to 

 be substituted for -368. -But for most purposes I should 

 imagine that either -37 or -SOS will serve. 



T. S. B.\RRETT. 



[Mr. Barrett's useful formula will be more readily remem- 

 berfid by observing that it approximates, when n becomes 

 very great, to (^-1-1)". This may be easily proved from 

 De Morgan's formula''' for 1x2x3 x . . . up to n — 

 |)i = ^/(2 7r h; h" ^ ~"+ ^li - 3,'io ,:?, + — A. C. llANy.-iKD.] 



ON TIIH RICD STARS XICAR TIIK CVC'rXLS NEBULA, 

 DETEt'TKl) HV DR. WOLF. 



To tlic Kditor of Knowledge. 



Sir, — The very remarkable photographs of Dr. Max 

 Wolf of the region about a Cygui, showing a large mass 

 of nebulous light, and your interesting article published 

 in the October number of KNowLumii;, led me to compare 

 the photographs with the charts of Argelander. Any red 

 star would at once bo recognizable, on account of the 

 difference between its actinic light and its brightness as 

 seen with the eye. All stars showing any such difi'erenco 

 were marked on the charts, and subsequently examined 

 with the telescope. The region is not abnormally rich in 



* .SVff \)e Morgan's " Uifferentinl Ciilcvilus," p. 312. 



known red stars. Only three fourth-type stars have 

 hitherto been detected in this region ; two others are 

 suspected to be of type IV. The red region of Cygnus lies 

 somewhat to the south of the x Cygni photograph, and all 

 the Wolf-Rayet stars, save one, arc also to the south. The 

 telescope used was a 17J-inch reflector, with a power of 70. 

 and where there was sufficient light the stars were 

 examined with the spectroscope, but no new fourth-type 

 stars were detected, though several new third-type stars 

 were discovered. The work was carried on during the 

 evenings of September 9th, 10th, 29th, October 1st, 2nd. 

 3rd, after I had first seen a print of Dr. Wolf's photograph 

 lent me by Mr. Ranyard, and in all 1.59 stars were 

 examined. The classification by colour is the same as 

 that used throughout the "Red Star Catalogue." The 

 following red stars were detected : — 



Red and orange red 23 



Pale orange red 38 



Slightly tinged with red 36 Total 97 



The remaining stars show nothing of interest, being 

 usually fainter than the 9-5 magnitude of Argelander. 

 Nine stars, however, are now only of the 11-0 magnitude, 

 or less, and several, if not all, will probably turn out to be 

 variable. The bomidaries of the smaller photograph are 

 in right ascension 20h. 11m. to 21h. Om. roughly, and 

 in declination + 39° to + 50". The number of stars in 

 Argelander is 3011. On page 191 of the "Red Star 

 Catalogue," amethod is given of finding the probablenumber 

 of red stars ; this is, number of stars x -0075. Multiply- 

 ing the number of stars in Argelander by this quantity, we 

 get 23, which accords exactly with the number of st-ars 

 actually observed. It would seem, therefore, that there is 

 no abnormal quantity of red stars in this region. In the 

 next place, the whole of the stars which showed any colour 

 were marked with red on the photographs to see if they 

 were distributed according to any rule. Only one soar, and 

 this pale orange red, was found in the nebula. Generally 

 speaking, the coloured stars seem to lie on the borders or 

 between the groups or streams of stars. A curious and 

 slightly curved line of four nearly equidistant coloured 

 stars extends from a Cygni to the preceding edge of the 

 photograph. Another curve of five stars is associated 

 with 7 Cygni. The north part of the photograph, especially 

 where the stars rapidly decrease in number, seems richer in 

 coloured stars, and on extending the sweeps yet further 

 north this fact is very obvious. As regards the minute 

 stars, I am inclined to believe that none are beyond the 

 range of the \1\ reflector. The minimum risihU- of the 17| 

 has been found to be 15 magnitude on Argclauder's scale, 

 and though only one field has been examined, yet from the 

 results it would seem that the smallest stars are within 

 the hmit of 15 magnitude. Only three red variable stars 

 known at the present time lie in this region, two of them 

 following a Cygni, and north of it ; still further to the 

 north and on the following side of the photograph, but 

 beyond it, lie three other variable stars. Finally, the 

 brighter stars in the immediate vicinity of the nebula 

 were examined with a powerful spectroscope, and seen to 

 be generally first-type, with strongly marked hydrogen 

 lines. It was thought possible that some of them might 

 show bright lines, as in the case of stars associated with 

 the nebula in Orion, but no such star was detected. The 

 nebula is visible in the 17|, as a faint haze. 



Towlaw, Darlington. T. E. Espin. 



[I hope in a future number to give a photo-etching of 

 the stars in the x Cygni region with :\Ir. Kspins new red 

 stars marked upon it. 1 should be glad if other observers 

 with large telescopes would compare the stars they 

 can see m this region with those shown in the plate 



