48 



SCIENCE. 



[N. S. Vol. XI. No. 263. 



salient mathematical problems which have 

 arisen in the study of the earth ; * and the 

 present review may hence be restricted to a 

 rapid resum6 of the less salient but perhaps 

 more recondite problems, and to the briefest 

 mention of problems already discussed. 



Adopting the convenient nomenclatui'e of 

 geologists, we may consider the earth as 

 made up of four parts, namely, the atmos- 

 phere ; the hydrosphere, the oceans ; the 

 lithosphere, or crust, and the nucleus. 

 Beginning with the first of these we are at 

 once struck by the fact that much greater 

 progress has been made during the century 

 in the investigation of the kinetic phe- 

 nomena of the atmosphere than in the 

 study of what may be called its static prop- 

 erties. Evidently, of course, the phenomena 

 of meteorology are essentially kinetic, but 

 it would seem that the questions of pres- 

 sure, temperature and mass distribution 

 of the atmosphere ought to be determined 

 with a close approximation from purely 

 statical considerations. This appears to 

 have been the view of Laplace, who was 

 the first to bring adequate knowledge to 

 bear upon such questions. He investigated 

 the terrestrial atmosphere as one might in- 

 vestigate the gaseous envelope of an unil- 

 luminated planet. f He reached the con- 

 clusion that the atmosphere is limited by 

 a lenticular-shaped surface of revolution 

 whose polar and equatorial diameters are 

 about 4.4 and 6.6 times the diameter of the 

 earth respectively, and whose volume is 

 about 155 times that of the earth. J If this 



*0n the Mathematical Theories of the Earth. 

 Vice-presidential address before Section of Astronomy 

 and Mathematics of the American Association for the 

 Advancement of Science, 1889. Proceedings of A. A. 

 A. S., for 1889. 



fMeoanique Celeste, Livre III., Chap. VII., and 

 Ltvre X., Chapts. I.-IV. 



X Laplace's equation to a meridian section of this 

 envelope is 



x~^ — ^0 ~ ' + Joa;%os'0 = 0, 

 where x = rja, r being the radius vector measured 



conclusion be true our atmosphere should 

 reach out to a distance of about 26,000 

 miles at the equator and to a distance of 

 about 17,000 miles at the poles. It does 

 not appear, however, that Laplace attempted 

 to assign the distribution of pressure and 

 density, and hence total mass of the atmos- 

 phere within this envelope ; and I am not 

 aware that any subsequent investigator has 

 published a satisfactory solution of this 

 apparently simple problem.* 

 from the center of the earth and a the mean radius of 

 the earth ; a is the ratio of centrifugal to gravitational 

 acceleration at the equator of the earth ; $ is geocen- 

 tric latitude, and x,, is the value of a; for </i = t / 2. 



The problem of the statical properties of the atmos- 

 phere may be stated in three equations, namely : 

 A2K+47ri:p — 2u2 = 0, 



dp = pdV, 



In these V is the potential at any point of the atmos- 

 phere; p, p, T being the pressure, density and tempera- 

 ture at the same point ; Tc is the gravitation constant ; 

 and u is the angular velocity of the earth. The above 

 equation of Laplace neglects the mass of the atmos- 

 phere in comparison with the mass of the rest of the 

 earth. An essential difficulty of the problem lies in 

 the unknown form of the function /(p, "). 



* I have sought a solution with a view especially 

 to determine the mass of atmosphere. A class of solu- 

 tions satisfying the mechanical conditions of the fol- 

 lowing assumptions has been worked out. Thus, as- 

 suming jj = cp"' , which includes the adiabatic relation, 

 p^cp'*', and the famous Laplacian relation, djjjdp 

 = 2cp ; and the law of Charles and Gay-Lussac, p 

 = CpT ; there results 



, \Qo/ ^0 So 



where Q = x-^ — Xg— ^ + ^ax^cos^f 



defined above ; Qo is the value of Q for x = 1 and ^ 

 = 7T j2; and po, Po, ''o are the values of p, p, r at the 

 same point (,t = 1, (6^77/2). 



"Using the adiabatic law the above formula for p 

 leads to a mass for the atmosphere of about l/1200th 

 of the entire mass of the earth. But since the adia- 

 batic law gives too low a pressure, density and tem. 

 perature gradient, this can only be regarded as an 

 upper limit to the mass of the atmosphere. A lower 

 limit of about 1/lOOOOOOth of the earth's mass is 

 found by assuming that the mass of the atmosphere 

 is equal to the mass of water or mercury which would 

 give an equivalent pressure at the earth's surface. 



Po 



(l> 



