July 14, 1892] 



NATURE 



249 



spouds), then on the surface of the Earth we may take, in 

 round numbers, 



g = CE/R«, or CE = gR^, 

 R denoting the radius of the Earth in cm, taken as 

 10® -r- ^TT, the quadrant being lo^ cm. 



With mean density p, the weight of the Earth, E, in g, is 

 given by 



E = ^irpRs, 

 so that 



^irRCp = g, 

 or 



Cp = ir X 10-9 ; 



so that p is known from C, and vice versd. 



For instance, with p = 5-5, and^ = 981, 

 C = 10-8 X 6-688. 



We are awaiting with great interest the quantitative 

 results of Mr. C. V. Boys, with his improved form of appa- 

 ratus ; but meanwhile we may take a mean density of 5-5, 

 the mean of Cornu's and Poynting's results, which is about 

 half the density of lead. It is very extraordinary that 

 this should agree so exactly with Newton's conjecture, 

 Principia, lib. iii., prop. x. :— " Unde cum terra communis 

 suprema quasi duplo gravior sit quam aqua, et paulo 

 inferius in fodinis quasi triple vel quadruple aut etiam 

 quintuple gravior reperiatur : verisimile est quod copia 

 materije totius in Terra quasi quintuple vel sextuple 

 major sit quam si tota ex aqua constaret ; prassertim cum 

 Terram quasi quintuple densiorem esse quam Jovem 

 jam ante estensum sit." 



5. A short numerical calculation will now give us the 

 weight of the Earth (Hamilton, "Lectures on Quater- 

 nions ") ; also of the Moon, Sun, &c. 



We assume that the Earth is a sphere, whose girth 

 is 40,000 kilometres, so that R, the radius of the Earth, 

 is 10' -=- iTT m (metres), and the volume, V, is ^rrR^ m', 

 while the weight, E, is pV t (metric tonnes of 1000 kg, 

 or 2205 lbs), where p — 5'5. 



Four-figure logarithms will suffice for our calculations ; 

 a nd now 



R = lo** X 6*366 m, 



I -081 



lo-i X 5-946 t, 

 or 6 X lo-i metric tonnes in round numbers. 



The weight of the Moon, M, generally taken as one- 

 80th of the Earth, will be 10^^ X 7"432 t. 



To determine S, the weight of the Sun, we employ 

 Kepler's Third Law, which gives 



S + E + M _ «^a3 

 E + M «'V»' 



where n, n denote the mean motions of the Sun and 

 Moon, and a, a' their mean distances from the Earth. 



Since M is insignificant compared with E, and E com- 

 pared with S, we may write this 



E «'V3' 

 where n'ln =13, the number of lunations in a year, and 

 aid! = 390, the ratio of the mean distances of the Sun and 

 the Moon, this being the ratio of 57' to 8"-8, the inverse 

 ratio of the parallaxes. 

 Now 



log a/a' = 2 -59 1 1 

 log (a/a')' = 77733 

 log («'/«)" = 2-2279 



log S/E = 5 '5454. S/E = 351.100; 

 so that the weight of the Sun is about 350,000 times the 

 weight of the Earth, or about 2 X 10-" t, or 2 X lo^^g. 

 To determine the value of G the acceleration of gravity 



NO. I 185, VOL. 46] 



on the surface of the Sun, compared with g, the value on 

 the surface of the Earth, we have 



G ^ S / diameter of earth y _ S^ fS^ \- 

 g E \ diameter of sun / E \96o/ ' 

 since the apparent semi-diameter of the Sun as seen from 

 the Earth is about 960", while the apparent semi-diameter 

 of the Earth as seen from the Sun, in other words the 

 solar parallax, is taken as 8"-8. 

 Now 



log 960 = 2 9823 

 log 8 -8 = -9445 

 log (960 -^ 8-8) = 20378 

 log (960 ^8-8)2 = 4-0756 

 log S/E = 5-5454 

 log G/g'= 1-4698, G/g= 29-49. 



6. According to Newton's Law of Universal Gravita- 

 tion, the operation of weighing out the quantity W in 

 the balance gives the same result wherever the operation 

 is carried out in the universe, assuming that the balance 

 and the body to be weighed are of ordinary moderate 

 dimensions. 



It is otherwise with the quantity denoted by P, because 

 the magnitude of the gravitation unit of force varies, being 

 proportional to the local value of ^. 



Suppose we write the first two equations 

 Pgs = iWz;2, Tgi = Wv, 

 and now put P^ = Q ; this is equivalent to taking a new 

 unit of force, i/^th part of the former unit ; this is an 

 invariable unit. 



Now our dynamical equations become 

 Qs = JWz/2, Q^ = Wv, 

 from which ^ has disappeared. 



The first suggestion of the change to this new absolute 

 unit of force is due to Gauss, who found the necessity of 

 it when comparing records of the Earth's magnetic force, 

 made at different parts of the Earth's surface, and all 

 expressed in the local gravitation unit. 



It is curious that this suggestion of an absolute unit of 

 force, the same for all the universe, did not originate with 

 the astronomers ; but Astronomy remains mere Kine- 

 matics until an accurate determination of the Gravitation 

 Constant has been made. 



On the F.P.S. (British foot-pound-second) system, this 

 absolute unit of force is called the poundal, a name due 

 to Prof. James Thomson ; so that 



Q5 = ^Wz'^ (foot-poundals), Q/ = Wz* (second-poundals). 



On the C.G.S. (Metric centimetre-gramme-second) 

 system, this absolute unit of force is called the dyne, the 

 centimetre-dyne of work being called the erg, and the 

 second-dyne of impulse being called the bole; and now 

 Q^ = ^-^v"- (ergs), Q^ - \\v (boles;. 



These absolute units are always employed in the state- 

 ment of dynamical results in Electricity and Astronomy, 

 where cosmopolitan interests are considered. 



7. The disappearance of^'-from the dynamical equations 

 is such a comfort to the algebraist, that he now makes a 

 new start ad initio in dynamics, and gives a new definition 

 of the absolute unit of force. 



He defines the poundal as the force which, acting on 

 a pound weight, makes the velocity grow one foot per 

 second every second ; and he defines the dyne as the 

 force which, acting on a gramme weight, makes the 

 velocity grow one centimetre per second every second ; 

 and now if W lbs. or g is acted upon by a force of Q 

 poundals or dynes, the acceleration a is given by 



a — Q/VV {celoes or spouds), . 

 and 



Q = Wa, 

 leading to the original equations 



Qj = JWz'^ (foot-poundals or ergs), 

 Q/ = W© (second-poundals or boles). 



