406 PHILOSOPHICAL TRANSACTIONS. [ANNO 1798. 



cause may perhaps act always, or commonly, in the same direction, and so make a 

 considerable error in the result. But yet, as the experiments were tried in various 

 weathers, and with considerable variety in the difference of temperature of the 

 weights and air, and with the arm resting at different distances from the sides of the 

 case, it seems very unlikely that this cause should act so uniformly in the same way, 

 as to make the error of the mean result nearly equal to the difference between this 

 and the extreme ; and therefore it seems very unlikely that the density of the earth 

 should differ from 5.48 by so much as T ' T of the whole. 



Another objection perhaps may be made to these experiments, namely, that it is 

 uncertain whether, in these small distances, the force of gravity follows exactly the 

 same law as in greater distances. There is no reason however to think that any 

 irregularity of this kind takes place, until the bodies come within the action of what 

 is called the attraction of cohesion, and which seems to extend only to very minute 

 distances. With a view to see whether the result could be affected by this attrac- 

 tion, I made the 9th, 10th, 11th, and 15th experiments, in which the balls were 

 made to rest as close to the sides of the case as they could ; but there is no differ- 

 ence to be depended on, between the results under that circumstance, and when 

 the balls are placed in any other part of the case. 



According to the experiments made by Dr. Maskelyne, on the attraction of the 

 hill Schehallien, the density of the earth is 44.* times that of water ; which differs 

 rather more from the preceding determination than I should have expected. But I 

 forbear entering into any consideration of which determination is most to be de- 

 pended on, till I have examined more carefully how much the preceding determina- 

 tion is affected by irregularities whose quantity I cannot measure. 



Appendix. — On the Attraction of the Mahogany Case on the Balls. 



The first thing is, to find the attraction of the rectangular plane ckfib (fig. 8,) on 

 the point a, placed in the line ac perpendicular to this plane. 



a* b z 



Let ac = a, ck = b, cb = x, and let 5 = w 1 , and , = = i/ 2 , then the 



attraction of the line bQ on a. in the direction ab, is = -v- — - ; and therefore, if 



ao -f- ap * 



cb flows, the fluxion of the attraction of the plane on the point a, in the direction cb, 



bx X — bw —bw — v 



"~ vV + x* x Va z + b 1 - r x i ^a 7 + x 1 " /Z~.~ t ~~" ^b*w z + a x ~~ ^\ + v q ' 



■V ° + a 



the variable part of the fluent of which is = — log. v + ^ 1 + t/ 2 , and therefore 

 the whole attraction is == log. ( c + a x , a °, a ) so that the attraction of the plane, 



v ac p/3 + a/3 ' 



in the direction cb, is found readily by logarithms, but I know no way of finding 



* The mean density of the earth by that experiment, has since been found to be nearly 5, or 5 times 

 that of water, by taking the real density of the hill, instead of one that was assumed below the truth. 

 See p. 420, vol. 14, of these abridgments. 



