124 



♦ KNO^A?'LEDGE * 



[April 2, 1888. 



of the earth.* From observations made on Mont Cenis by 

 this plan, Plana and C'arlini deduced a v.alue of 4-950 for the 

 earth's mean density. 



The converse of this method has also been used. If the 

 oscillations of a pendulum at the S3a-level be compared with 

 those of a pendulum at a great depth below that level (as in 

 a deep mine), we can compare the mass of the whole earth 

 with that of an outer shell limited by an imaginary spheroidal 

 surface concentric with the surface of the earth, and passing 

 through the undergi-ound station. For this shell, supposing 

 it homogeneous, would exert no attraction at all at the lower 

 station, and its non-homogeneity can be taken into account. 

 The pendulum at the lower station is attracted only by the 

 mass within the imagined spheroidal surface, with such 

 corrections as may be due to the irregularities of the 

 structure of the shell exterior to it, and more particularly 

 to peculiarities in the neighbourhood of the mine where the 

 experiment is tried. If the earth were of uniform density 

 the attraction at the lower station would obviously be less 

 than that at the outer, in the same degree that its distance 

 from the centre is less ; for masses of the complete sphere 

 and of the sphere within the lower station are proportioned 

 as the cubes of their radii, while gravity .at the two stations 

 would be as these masses directly aud inversely as the 



the centre, the attraction at the lower station may be equal 

 to or greater than that at the tipper. Apart, then, from 

 considerations depending on the contigitration of the mine 

 and the structure of the strata through which it has been 

 dug,* the interpretation of the observed difference of 



* The effect o£ the removal of large quantities of material from 

 below tlie surface is to increase the attraction on the pendulum at. 

 the bottom of tlie mine. It will suffice, in illustration of this, to 

 consider the earth regarded as a homogeneous globe, and the space 

 dug out as of some regular figure, as a cylinder. Tlie student will 

 readily infer (which is ;iK I wish here to indicate) the nature of the 

 general considerations which have to be taken into account in deal- 

 ing with such problems : — 



Let B, fig. 2, be a mine of any figure, with, however, a horizontal 

 base. Take the spherical shell A 8 C, of which the inner surface 

 coincides with the base of the mine. Then the attraction at the 

 bottom of the mine is that due to the mass of the sphere within the 

 shell, plus the attraction of the incomplete shell ; .and i he incom- 

 plete shell attracts the particle towards the earth's centre with a 

 force precisely equal to the repulsion due to a mass fillini:;- the space 

 B. For if the attraction exerted by the incompleie shell be A, and 

 that exerted by the portion 1! (supposed filled) be B, then we know 

 that 



A + E = 

 or 



A = - B. 



Teace, instead of considering the incomplete shell A, we need 



FiQ. 2. -Illustrating the Mine llethcd of Weigliing the Earth. 



Fig. 3.— Illustrating the Determination of Gravity at the Bottom of a Mine. 



squares of those radii ; hence, considering both the masses 

 and the distance, the attractions at the two stations would 

 be inversely as the distances of these stations from the 

 earth's centre. But if the earth's density increases towards 



* The time of oscillation of a pendulum oE given length, I, is 

 inversely proportional to Vy_ where ff represents the force of gravity 

 as measured by the velocity it can generate in a unit of time. 

 Hence, if //, ;/' represent gravity at tlie sea-level and at height /( 

 above that level respectively, and f, i' are the respective periods of 

 oscillation as calculated for stations so situated resi)ectively (the 

 attraction of the mountain itself being neglected), we should have, 

 approximately, j 



: : r : r-\-U 

 {r being the earth's radius). For, regarding the earth as a globe 

 made up of concentric layers, each of uniform density, the earth's 

 whole mass may be supposed at the centre, and gravity therefore 

 inversely as the squares of the distances from the centre — that i.s, 

 (J :. <?'::(?• + 70-' : ■>•-. 



But if calculations based on the known configuration and struc- 

 ture of the mountain indicate that its mass would produce an 

 attraction /on a body at its summit, and t" be the observed time of 

 oscillation at the mountain's summit. 



Thus, if the times of oscillation t and t'' be accurately oompa'-ed, 

 while t' is computed, we obtain the ratio of t' to t", or V,,' j^j to 

 -./^■^ whence the ratio of / to (j' and thence to g is obtained ; and 

 thus the mass of the earth may be compared directly with that of 

 the mountain. Of course, though the principle of the method is 

 thus indicated, the actual computations are by no means so simple. 



only consider the repulsion exerted by a mass filling B, every 

 particle of which is to b ; supposed to repel with a force exactly 

 equal to that with which it would in reality attr.^ct. 



Let us suppose our mine to be cylindrical, tig. .S, representing a 

 section through the axis and the sbaft S coincident with the pro- 

 longation of the axis. A pendulum is swinging at Q and another at 

 P ill Q S produced. The shaft is supposed so narrow that we need 

 not consider it, and, for convenience, we suppose its length equal to 

 half the depth of the mine: — 



Let r = earth's radius 

 2 a = mine's depth (Q P) 

 a = length of shaft (Q S) 

 h = radius of cylinder (B P) 

 Then the attraction at Q is equivalent to the attraction of a 

 sphere of radius r, diminished by the attraction of the cylinder A U, 

 that is (from the known value of the last-named attraction — 



= *--.f - 2 IT p |a - s/ia^ + b' + ■Jd^^n/'] . (») 



(p being the density of the supposed homogens-.us sphere). 



Ag.ain, the attractiou at P is e.juivalent to the altr.acti n of a 

 sphere of radius r — 2a';, increased by the repithioii of the cyliLder 

 A D, that is- 



The excess of the attraction at Q over that at P— that is, the 

 difference of the expressions (a) and (;8) — is 



I 





-I- b- — 2a — h 



'S 



r 



= 2k p ' Via- -(•*-- *| - 



i TT pa. 



(7) 



and, according as this expression is positive, zero, or negative, the 



