Royal Society. 515 



ellipsoid passing through the point attracted, and having the 

 same foci as another ellipsoid (passing through the same point) 

 whose corresponding semiaxis is «j. Hence, for an interior 

 point, we have 



A=4..^(i-.^)v/(i-oj; ^(i_,j;5V-.>v) ' 



B = 47ryi/(l-e2)-/(l-e'2) {' 1^ r 



^ ^ .'o (l-eV)4(l-e'V)* 



C=47r^^/(l-e2)^(l_e'-2) T 



Jo (1- 



u^du 



(l-eV)i(l-e'V)^ 

 These are the known expressions for the attraction of an elHp- 

 soid on a point within it. Equations (5) agree with the expres- 

 sions given in the supplement to liv. v. of Pontecoulant^s Theorie 

 Analytique du Systeme du Monde, where they are found by direct 

 integration by a method discovered by Poisson. They may also 

 be readily deduced from equations (6) by Ivory's theorem. Or, on 

 the other hand, by a comparison of them, after reducing the limits 



of the integrals to and 1 , by substituting -p v for u, with 



equation (6), Ivory's theorem may be readily demonstrated. 



LXXXI. Proceedings of Learned Societies. 



ROYAL SOCIETY. 



[Continued from p. 441.] 



March 30, 1854.— Thomas Bell, Esq., V.P., in the Chair. 



T^HE following paper was read : — -'On the Structure and Affini- 



-■■ ties of Triyonocarpon (a fossil fruit of the Coal-measures). 



By Joseph D. Hooker, M.D.. F.R.S. 



Having been for some time engaged in examining the structure 

 and affinities of some fossil fruits of the coal formation, included 

 under the name Trigonocarpon, and the progress which I am enabled 

 to make being extremely slow (owing to the difficulty of procuring 

 good specimens), I am induced to lay before the Royal Society such 

 results as I have arrived at, for publication in their Proceedings (if 

 thought worthy of that honour). The details and illustrations of 

 the subject will, when complete, be offered to the Geological Society 

 of London. 



My attention has for many years been directed to the genus Tri- 

 gonocarpon ; as, from the period of my earliest acquaintance with 

 the flora cf the carboniferous epoch, I have felt assured, that bota- 

 nically, this was the most interesting and important fossil which it 

 contained in any great abundance, and that until the affinities of this 

 were determined, the real nature of the flora in question could never 

 be regarded as even approximately ascertained. 

 2 M 2 



