132 



ON THE LAWS OP 



a table of the values of the density at different points on the 

 surface of the plate, calculated by means of the formula (29), 

 together with the corresponding values found from experiment. 



We thus see that the differences between the calculated and 

 observed densities are trifling ; and moreover, that the observed 

 are all something smaller than the calculated ones, which it is 

 evident ought to be the case, since the latter have been deter- 

 mined by considering the thickness of the plate as infinitely 

 small, and consequently they will be somewhat greater than 

 when this thickness is a finite quantity, as it necessarily was in 

 Coulomb's experiments. 



It has already been remarked that the method given in the 

 second article is applicable to any ellipsoid whatever, whose 

 axes are a, 5, c. In fact, if we suppose that x, y, z are the 

 co-ordinates of a point p within it, and #', y ', z those of any 

 element dv of its volume, and afterwards make 



x = a . cos 0, y = b . sin 6 cos OT, z c . sin 6 sin w, 

 x = a . cos ff , y = b . sin & cos tn-', z c . sin & sin <*/, 



we shall readily obtain by substitution, 



F= abcfp . r z dr'de'd<v' sin &. (Xr 2 - 2/m-' + vr'^ ; 



the limits of the integrals being the same as before (Art. 2), and 



X = a 2 cos 2 + 2 sin tf 2 cos *? + c* sin & sin ^ 2 , 



/i = a 2 cos 6 cos &+ b* sin 6 sin ^cos w cos r'+ c 2 sin 6 sin 0'sin CT sin OT', 



v = a 2 cos &* + tf sin 0' 2 cos -cr' 2 + c 2 sin 0"* sin ' 2 . 



