Mav 8, 1SS5.J 



• KNOWLEDGE 



403 



ipiir iHatt)fmatiral Column. 



MATHEilATICS OF METEORIC ASTRONOMY. 

 By Richard A. Pkoctor, 



{ Continued from p. 3;!0.) 



IT will be well uext to consider the case of a planet regarded at 

 first as actin? alone on a meteor. Of conrse no meteoric body 

 can erer be dra«-n towards a planet under the iulluence of the 

 planet's attraction solely, since each such body entering our sol.ir 

 system is under the chief attraction of the sun, whose mass exceeds 

 the combined mass of all the rest of the system nearly 730 times, 

 and the mass of Jupiter the largest planet 1,048 times. Still we 

 shall tind the advantage of ini|uiring what velocities Jupiter and 

 other planets could severally impart to meteoric bodies approaching 

 them nnder their sole attraction. 



Let US begin with the earth herself. In her case we know the 

 actual force we have to deal with from direct evidence. We know 

 that the earth's mass acting on bodies at her surface, 3,9G0 miles 

 from the centre (where all the mass may, as Newton long since 

 showed, be supposed to be gathered) imparts in one second a 

 velocity of 322 feet per second. It will bo well, however, to use 

 the mile as the unit of length, in which case terrestrial gravity at 

 the earth's surface is represented by 322 -i- 5,280, an<l the constant 

 /I for this case, since it represents the force at a distance of one 

 mile, is given by the equation 



^ = .^x(3960r 

 = ?r? X 33 X 396 = 322 x 3 x 99 

 = 95,634. 



[It will be noticed that this determination of /i for the earth is 

 independent, while the determination of /z for the sun depended 

 only on the determination of the sun's distance as 92,600,000 miles. 

 Thus we may regard these two determinations as involving an 

 independent measurement of the sun's mass, which 



earth's mass 



II as determined for the sun 

 "/i as determined for the earth 



31,298,800, 000 

 "" 95,631 



= 327,300, calling the earth's mass unity.] 



To determine, then, the velocity of a body approaching the earth 

 nnder her sole attraction from a very great distance, at any given 

 distance x, we have 



j_ 2x 95,634 ^ 191268 



X X 



Thus, if we require the velocity of snch a body at the earth's sur- 

 face we have 



^,^191268^48.3 

 3960 



r = 6-95 approximately. 



Thus a body drawn from an infinite distance to the earth's surface 

 nnder her attraction only, would reach her surface with a velocity 

 of nearly 7 miles per second, and a body would have to be shot 

 from the earth's surface with this velocity to pass away for ever 

 from her, if she were the only orb in existence. 



Again, at the moon's mean distance, 238,818 miles, we should 

 have - 3S 



, 191268 



and 



238818 

 r = -895 



= 0-8009 



This is the velocity with which a body approaching the earth 

 nnder her sole attraction from rest at an infinite distance would 

 cross the moon's orbit. The moon's mean velocity, apart from the 

 strn's influence, which slightly diminishes the earth's influence on 

 her, would be 



= -895 X V2'= -6328 (miles per second) 



= 227808 miles per hour. 



But now, before leaving the earth, it ma.v be interesting to 

 inquire what are the velocities which the earth can add to velo- 

 cities already imparted by the san to bodies travelling up to 

 the earth on their way towards him. In order to get the maximum 

 effect which the earth may be supposed to impart in this way, and 



also to simplify the problem. I suppose the earth at rest, and the 

 motion of tho approaching body to take place in a straight line : — 



A 



I 



I I 



Fig. 



Let then tho body move alon'^ the straight lino A PE S, Fig. 2, 

 towards tho earth at E and tho sun at S. I'utPS = .r, ES = R (aa 

 before). Then, dealing with this problem as wo dealt with the 

 simpler case of the sun alone attracting, wo get, writing /i for tho 

 solar constant /i' for the terrestrial constant of force, the equation 



d',r_ ft /t' 



<lT-~ ? (x-K)-' 

 And treating this as before, we get 



\dfJ X (x-K) 



(x-K) 



In the case of a body starting from rest at an infinite distance, wo 

 have C=0. Wherefore, in this case 



X T — R 



Xow put (x-R) = radius of earth = 3960; and a; = 92,600,000. 

 Then, substituting for /i and fi' their values as already determined, 

 we have, for r the velocity of the body as it reaches the earth's sur- 

 face, the equation 



670x92.600,000 191,268 



and 



)2,li00,000 3900 



4-3; 



= 676 + 48-3 = 

 r = 2tJ-9 



So that the increase of velocity, even in the case of a body which, 

 having been drawn towards the sun from rest at a very great dis- 

 tance, comes eventually to the very surface of the earth, could not 

 be more than 9-lOths (26-9 — 26) of a mile per second, even if the 

 earth were airless and so the body came with unimpaired speed to 

 the surface. In every case, then, the influence of the earth on 

 bodies reaching her from outside with sun-imparted velocities must 

 be comparatively insignificant — unless these bodies come upon her 

 from behind (speaking witli reference to her own motion), when, 

 their relative velocities being diminished by her ovvn velocity, may 

 be reduced to 7i miles per second. In such cases terrestrial 

 attraction might quite largely aflect the apparent directions of 

 meteoric motion. However, these cases are comjiaratively in- 

 frequent. 



(To be continued.) 



Anthropological Institote. — On April 28 (Francis Galton, Esq., 

 P.R.S., President, in the chair) Mr. A. L. Lewis read a paper on 

 the past and present condition of certain rude stone monuments 

 in Westmoreland. A little to the south of the village of Shap are 

 the remains of some very extensive rude stone monuments, to 

 which allusion was made by Camden in the fifteenth century, and 

 by Dr. Stukeley in the middle of the last century, and a circle is 

 said to have been destroyed when the railway was made. The most 

 interesting monument in this neighbourhood is .situated at a place 

 called Gunnerskeld. two or three miles to the north of the village, 

 and consists of two irregular, concentric, slightly oval rings, about 

 fifty and a hundred feet in diameter respectively, the longest dia- 

 meters being from north to south. A paper by Admiral F. S. 

 Tremlett on " Quadrilateral Constructions " near Carn.ic was 

 read, which described certain enclosures explored by the late Mr. 

 James Miln. In each case the boundary walls are formed of coarse 

 undressed stones, put together without any kind of cement, and 

 having built up in them a series of small menliers. They also con- 

 tained beehive structures for cremation, reddened and become friable 

 from the effects of great heat; it would appear that the crema- 

 tion had been perfect, as not a particle of calcined bone was found 

 in either of the enclosures. A paper by M. Jean L'Heureux, on 

 tho " Kekip-Sesoators, or Ancient Sacrificial Stone of the North- 

 West Tribes of Canada," was read. The stone, which consists of 

 a roughly hewn quartz, or boulder, about 15 in. high, and 14 in. in 

 diameter, is placed on the summit of a pyramidal mound, com- 

 manding an extensive view of both the Red Deer and Bow River 

 valleys. In cases of public or private calamity, or when a special 

 blessing is sought, a .solitary warrior, after keeping vigil on the 

 top of tho mount from sunset till the rising of the morning star, 

 then lays a finger of his left hand on the top of the stone and 

 cuts it off. Amongst the Blackfeet these self-inflicted wounds 

 ranked equal to those received in battle, and are always mentioned 

 first in the public recital of the warrior's great deeds. 



