726 



NATURE 



[November 17, 192. 



the spherical formations hanging <lownwards with 

 clear cut edges. If the photograph l)e turned upside 

 down the ju.r,. nranrc is that of the tops of cumulus 

 clouds as .m aeroplane alK)ve them. 



Just as t A V tops of cumulus clouds are due 



to the ascent of warm moist air into cooler air above, 

 so the Klobular formation of the festoon-cloud must 

 he caused by the descent of warm moist air into an 

 uiulcriying cooler stratum. This inversion of tem- 

 perature is generally indicative of bad weather, and 

 this was corroborated by the weather experienced at 

 and after the time the photograph was taken. 



William J. S. Lockyer. 



Norman Lockyer Observatory, 

 Sidmouth, South Devon. 



The Tides. 



The great importance of the subject is my excuse 

 for troubling you once more, very briefly, regarding 

 it. In Nature of July 21, I stated that, according 

 to the present tidal theory, the tidal forces, and 

 consequently ,the tides, would be just the same for 

 a sea-depth of about 4000 miles as for the actual 

 sea-depth of about 2 miles ; and, in the same issue, 

 your reviewer, " The Writer of the Note," agrees that 

 this is true, or, in his own words, " that the differential 

 motion of the oceans is determined by the vectorial 

 excess of the forces at the earth's surface over those 

 at its centre " ; which appears to ignore entirely the 

 depth of the ocean as a factor determining the height 

 of the tides. 



The theoretical cause of the tides is the diflference 

 of the attractions of thesun and moon at the earth's 

 surface and centre. This difference in the case of the 

 moon is more than twice as great as in the case of 

 the sun ; therefore, the lunar tide is more than twice 

 as great as the solar tide. Similarly, if the earth 

 were expanded into a hollow, spherical crust of ten 

 times its present diameter, with its water-covered 

 surface nearest to the moon at the same distance as 

 now, and the moon's period of revolution also remain- 

 ing the same, then the lunar tide-raising force, and 

 consequently the tide, would be about twelve times 

 as great as now. This is the teaching of the present 

 tidal theory ; but is it the teaching of practical 

 mechanics and common sense ? Why shoidd the 

 mere expansion of the earth cause a ten, or twenty, 

 or a hundred time? greater tide upon its surface, the 

 distance of that surface from the moon, as well as 

 the masses of the earth and moon, remaining the same 

 as before the expansion ? 



Surely this is a question well worthy of discussion ; 

 and surely some of your readers are sufficiently in- 

 terested and open-minded to express some opinion or 

 argument regarding it. Evan McLennan. 



Corvallis, Oregon, U.S.A., September 3. 



Mr. McLennan's words " and consequently the 

 tides " are not in accordance with dynamics and are 

 not implied in the passage he quotes from my previous 

 note. If the earth were all water the direct tide- 

 generating forces within two miles of its surface would 

 be the same as in an ocean of depth only two miles. 

 These tidal forces are usually represented by reference 

 to the " equilibrium tide," that is, by stating what 

 the outer surface of the oceans would be if the water 

 had lost its inertia without losing its gravitational 

 properties. This outer surface would be the same in 

 the two cases mentioned. The necessary continual 

 adjustment of water, however, would be quite different 

 in the two cases ; in the first case the water within 

 two miles of the surface would be largely raised and 

 lowered by that beneath, while in the second case the 

 water would move mainly in a horizontal direction. 



NO. 2820, VOL. I 12] 



But owing to the actual •"•••-••■ - ' ♦»«e water the outer 

 surface olthe ocean W' \- different in the 



two ca.ses, so that the a--,^,..^- .... jiy docs not ignore 

 the (If pth of the ocean as a factor determining the 

 height of the tidts*. 



The expansion of the solid earth, with an 

 in water sufficient to conserve the depth of tlf 

 would magnify the tides l)ecause f ;.. 



forces at the earth's surface over th !i<- 



would expand with the earth's radius. Mr. Mci^nnau 

 apparently finds this result of the gravitational theory 

 repugnant to his common sense. 



The Writer of the previous Notes. 



Stirling's Theorem. 



In connexion with the recent letters published in 

 Nature on Stirling's Theorem, I beg to say that in a 

 paper accepted for publication by the Academy of 

 Zagreb on July 13, and now in print, I proved in 

 quite an elementary manner the formula 



n! = ^2*- •(»+«)'•■*'*• <J~^"^''^ 

 a =0'2i 13249 or 0788675 1 , 

 which coincides with the results published by Mr. 

 James Henderson in Nature of July 21, p. 97, formula 

 (3) . The error was found to be of the order of i /72 J^n' 

 of the calculated value, where 1/72 ^'3 is equal to 

 000801875 in Mr. Henderson's results. The formula 

 may also be written 



and the log p determined once for all. (For 

 a =--0-2113249, we have log /» = 0-5244599.) The work 

 of calculation is then by no means greater than in 

 using Stirling's or Mr. H. E. Soper's formula though 

 the approximation is far closer. I think the doubt 

 inferred by Mr. G. J. Lidstone in Nature of August 

 25, p. 283, on the usefulness of the formulae under 

 discussion is not valid so far as the present one is con- 

 cerned. For sufficiently large values of n, depending 

 on the number of decimals of the tables, the result 

 calculated from the above formula is not worse than 

 that furnished by any other more complicated formula. 



Stanko Hondl. 

 Zagreb, Croatia, SHS-State, 

 October 7. 



Prof. Hondl's simplified form of my best first 

 approximation to the value of «! follows at once 

 from the fact that (6 - c) = J in my letter in Nature of 

 July 21. [6 is F*rof. Hondl's a.] The constant p in 



n\-pi ) IS s 2«-^ 



We have now three approximations involving this 

 type of expression where the index of the power is 



(3) 



J^I^J^^!L±iy^\Forsyth]. 



It is interesting to note the increase in accuracy as 

 we proceed from (i) to (3). The errors are i/24n, 

 I /i 25*1*, and 1/240M* respectively. Of approximations 

 of this type Forsyth's is by far the most accurate, but 

 for logarithmic calculation it is rather more laborious. 



James Henderson. 

 Biometric Laborator\', 

 University College, London. 



