354 Royal Society. 



occur, they are sometimes of different kinds j cyclades and paludinre 

 are most plentiful. Anodons occur in it at Owthorne, but I did not 

 find them elsewhere. 



" The quadrupedal remains which have been found in this lacus- 

 trine formation, belong principally to deer. Bones of oxen likewise 

 occur in it. Of deer, at least, three species have been discovered in 

 the peat and clay ; the great Irish elk (C. giganteus), the red deer 

 (C. elaphus), and the fallow deer (C. dama). A doubtful skull, 

 (found at Owthorne,) in the possession of the Yorkshire Philosophi- 

 cal Society, has some resemblance to the cranium of the chamois 

 goat." 



(To be continued.) 



LVIL Proceedings of Learned Societies. 



ROYAL SOCIETY. 



Feb. 3. A PAPER was read, entitled, " On the Lunar Theory." 

 -* Communicated by the Rev. Dr. Lardner. 



The subject treated of in this paper is introduced by a review of 

 the labours of Clairault, Euler, D'Alembert, and Thomas Simpson. 

 The theories of these eminent men, the author remarks, were very 

 deficient in accuracy, and were not at all adequate, without correc- 

 tion from observation, to the construction of tables. They could 

 serve only to point out the arguments of the equations, and not all 

 even of these. The inequalities of the moon's motion are investigated 

 by approximating processes, which lead to results more or less ac- 

 curate, according as the approximations are carried to a greater or 

 less extent. The writers above mentioned had contented themselves 

 with short and easy approximations ; and though they had accom- 

 plished much, had yet left much more to be done. Subsequently 

 to these, Mayer published an elaborate theory of the moon ; but his 

 coefficients required much correction, the results of his computations 

 being in some cases found to differ very widely from observation. 

 A much greater degree of accuracy was attained by Laplace, who 

 bestowed particular attention on the influence of minute quantities 

 in every part of his theory. In the present paper the author has 

 endeavoured to introduce further improvements in the lunar theory, 

 by carrying the approximations considerably further than has hitherto 

 been done. 



In the solutions of the problem given by former mathematicians, 

 the chief obstacle to the attainment of accuracy was the extreme 

 length and labour of the necessary computations. Another object, 

 therefore, which the author has had in view, is to facilitate these 

 computations, and render them less laborious. This he endeavours 

 to effect by the employment of certain artifices, by which the multi- 

 plicity of small terms will, with their co-efficients, be reduced within 

 a practicable compass, and their numerical computation rendered 

 less appalling. 



The 



