MEAN DENSITY OF THE EARTH. 641 



the specific gravity was ascertained from some metal taken from the 

 same crucible from which the weights were cast. The relative position 

 of the weights and pendulum balls during the observations was an im- 

 portant factor, as was also the radius vector of each of the balls, and 

 they were most carefully investigated. It was found that after the first 

 few days there was no appreciable change in the length of the wire rope 

 supporting the weights, so that the employment of the mean relative 

 position could not introduce dangerous errors. The length of the 

 pendulum balls from the knife-edge, the principal element entering into 

 the computation of the moments, was measured by a standard catheto- 

 meter belonging to the Prussian commission of standards. The dis- 

 tance of the knife edge from the scale was ascertained by means of 

 direct measurement and by triangulation with results quite harmonious. 

 In the determination of all the constants the effort was made, and suc- 

 cessfully it is believed, to keep the error within one unit in the third 

 decimal place. 



Just here it might be mentioned that,* taking every possible element 

 into consideration, the resulting moment of the pendulum with the balls 

 for both positions of the weights was 10.3106^5^, while without the balls 

 it was 4.3578^, in which d represents the specific gravity of the weights. 



DISCUSSION OF THE OBSERVATIONS AND RESULTS. 



The observations consisted of the determination of the duration of 

 oscillation and the equilibrium position of the pendulum, for which a 

 series of fourteen observations were made with the balls on the pendu 

 lum and twelve with the balls removed. They were begun with a large 

 amplitude and continued until the arc passed over was only six or ten 

 minutes. In both cases a continuous decrease in the time showed it- 

 self, which could not be explained by the usual reduction to the 

 smallest swings. Hence the time was taken for two double swings 

 when the amplitude was at an average value, and two or more times 

 taken on each side of this value. These observations were adjusted 

 graphically, a method which answered in this case in consequence of 

 the regularity in the values. The theoretical time was compared with 

 the observed, and the discrepancy reduced to a minimum. Knowing 

 the equilibrium position of the pendulum, and the distance of the ball 

 from this point, at each elongation a simple subtraction gave the dis- 

 tance of the ball from the attracting body. 



The formula used in the final computation was : 



<7 = a constant JoTtR l + ff— ( nO'— f j cos* ^ 

 in which J is the desired density, R the polar radius, a the ratio of the 

 equatorial centrifugal force to the polar/^ .^ a^ ) ^ ^^^ elliptic! ty of the 



earth ( onoTr V^*^ ^ ^^'^ geographicall atitude (52^ 23"). This gave 

 H. Mis. 142 41 



