April 28, 1893.J 



SCIENCE. 



227 



what the lower part of the river has already lost: they tell us 

 what it has been, while it foretells what they shall come to be. 



Of course, while only the maps are before me, and the Osage is 

 a long thousand miles away, I do not wish to assert that this 

 sketch of its history is demonstrably true ; although I am strongly 

 persuaded that an examination of the region on the ground would 

 discover evidence confirmatory of it. The upland is built of 

 nearly horizontal Paleozoic rocks. If they had stood at their 

 present height above the sea. ever since the date of their 

 deposition, they would now be worn down close to sea- 

 level, without retaining any distinct relief. Their narrow val- 

 leys show that this supposition is out of the question. The roll- 

 ing upland in which the narrow valleys are incised is itself a 

 surface of denudation ; and as its reliefs are faint, with long 

 gentle slopes and broad open valleys, beneath whose floor the 

 narrow deeper valleys are incised, I am driven to the belief that 

 the upland was for a long time a lowland, and that its gentle 

 eminences are merely the remnants of a once higher mass. The 

 dates at which this older denudation was carried on, and the 

 later date at which the uplift to its new altitude was given, are 

 not well determined; although from analogy with more eastern 

 parts of the country, where the dates of such changes have been 

 better made out, I am inclined to say that the Missouri upland 

 was a lowland well into Tertiary time; and that the new trenches 

 of the Osage and its neighbors were begun in consequence of an 

 uplift somewhere about the close of Tertiary time. 



These are suggestions rather than conclusions ; but they still 

 serve to illustrate the incentive to geographical study that the 

 topographic maps supply. We all knew that there was a fertile 

 field for study in our home geography; every one in his own dis- 

 trict enjoyed cultivating his patch of the field ; but now through 

 the publication of these maps, it is as if the whole field was 

 opening to all of us; and a rich geographical product is promised 

 to all who enter it. 



SUN-HEAT AND ORBITAL ECCENTRICITY. 



BY ELLEN HATES, WBLLESLBT, MASS. 



The reader of Sir Robert Ball's important work, " The Cause 

 of an Ice Age," needs no reminder that its argument rests upon a 

 foundation of theoretical astronomy. To secure the essentials of 

 the discussion one must read between the lines. It is the object 

 of the present paper to select and arrange a few of the more sim- 

 ple inter-linear readings, in the hope that they may be serviceable 

 in that borderland where astronomy, geology, and meteorology 

 have each a claim. 



1. " There can be no doubt that when the eccentricity is at its 

 highest point the earth is, on the whole, rather nearer the sun, be- 

 cause, while the major axis of the ellipse is unaltered, the minor 

 axis is least." ("The Cause of an Ice Age," p. 79). This is equiv- 

 alent to saying that the mean distance of the earth from the sun 

 is a function of the eccentricity of the earth's orbit, and is, more- 

 over, such a function that when the eccentricity is a maximum 

 the function is a minimum. The mean or average length of the 

 radius-vector of an ellipse depends on the law assumed in regard 

 to its variation. From the standpoint of geometry, disregarding 

 kinematical and dynamical considerations, the simplest assump- 

 tion is, that the vectorial angle is the fundamental variable. If 

 the equation to the ellipse be written 



^ _ g (1 _ e^) 



1 + ecose 



and /•■ be the mean length of the radius- vector, we may easily 

 show that 





r dti = a » 1 — «2. 



(1) 



But in any investigation dealing with the amount of light or 

 heat received by the earth a different assumption should be made ; 

 for it is clear that if the earth moves most slowly when in aphe- 

 lion the effect is the same as if it were, on the whole, farther 

 away from the sun. Assuming that the time is the fundamental 

 variable and that the radius-vector sweeps over equal areas in 



equal times, we may find the average of the radii-vectores cor- 

 responding to the successive equal time-intervals. Consider a 

 point moving in a circle whose centre is one focus of the ellipse. 

 Let its areal velocity be equal to that of the point describing the 

 ellipse, and suppose that when the radius-vector of the ellipse 

 has swept through 180°, the radius, r^, of the circle has swept 

 through the same angle. Then 



" dt 



de 



= 3C: 



t^ l-e= 



where 3 T is the periodic time. Integrating between the limits 

 0° and 180" 



V 1-, 



= a/l- 



(3) 



r„ is thus a minimum when e is a maximum, and vice versa. The 

 value r,, in (2) is greater than the value r' in (1), as we might 

 have known in advance by simply comparing the two assump- 

 tions respecting the law of variation of r. 

 Developing the factor 



vt: 



V 4 33 / 



The present eccentricity of the earth's orbit is 0.01678. Accord- 

 ing to Leverrier it cannot exceed 0.077747. To take r^ = a, the 

 average of a (1 -|- e) and a (1 — e), that is, of the aphelion and peri- 

 helion distances, is therefore a close approximation to the mean 

 value obtained with the assumptions above made. Laplace, in 

 stating Kepler's third law, says, "The squares of their times of 

 revolution are as the cubes of the transverse axes of their ellipses." 

 {Mec. Cel., II., i., § 3). He uses the term "mean distance" in 

 speaking of the satellites of Jupiter and Saturn, but not in such a 

 way as to indicate that he meant the semi-major axis. Gauss, 

 in his first mention of the semi-major axis, says, "Hinc semi- 

 axis major, qui etiam distantia media vocatur, fit = — — — ". 



1 — ee 



("Theoria Motus," p. 4). Similarly, Sir John Herschel uses the 

 terms "mean distance" and "semi-major axis" as interchange- 

 able. 



2. "The total quantity of heat which the earth receives during 

 each complete revolution will be inversely proportional to the 

 minor-axis of the ellipse." (p. 79). Let dhhe the heat-increment 

 received in the time dt, and /^ the rate of variation of heat at a 

 unit's distance. Then, since the quantity of heat received varies 

 directly as the time and follows the law of the inverse square, 



dh = u — . 

 But from Kepler's second law, 



r^ = 2c, or — = — . 



dt dt de 



add _ fin 



~2c ~ Tc 



From this it appears that the quantity of heat received in passing 

 from one end of the major-axis around to the other varies in- 

 versely as the areal velocity. But 



Hence 



h 



-/■ 



(3) 



„ 7ra2 t^ 1 — e^ 



3c = , 



T 



and since the length of the year is constant and the major-axis is 

 constant, the areal velocity is to be viewed as a function of e alone. 

 Suppose e becomes e' and let c' denote the new value of the areal 



and therefore 7; : h' :: c : c. But 



velocity. Then h' = ^'^, 



■ a h 



;: 6 : 6' ; hence 7i 

 for 2 c in (3), 



%■ 



b' : b. Again, if we substitute 



h = liJLl = ^^ 



■77 a b c,s / 1 _ gs 



(4) 



Hence the amount of heat received in one year is the same that 

 would be received if the earth were to move for a year in a circle 



whose radius is a V' 1 — e=. 



