162 Mr. Ivory on such Functions as can he 



that subordinate part of the first theory which is well demon- 

 strated, and about which there is no dispute. Thus we 

 have the direct testimony of Laplace that his method was ori- 

 ginally extended beyond its just bounds. On the contrary 

 M. Poisson has always contended for the exactness of the 

 theory in its fullest extent. He undertook to clear it from all 

 objections by giving an unexceptionable demonstration of it, 

 first in the nineteenth cahier oi the Journal dc V Ecolc Poly- 

 technique, published in 1823, next in the additions to the 

 Conn. desTenis for 1829, and very lately in his Theory of Heat, 

 Prof Airy also turned his attention to this subject in the 

 Cambridge Transactions for 1826. He thinks that the ge- 

 neral theory of Laplace is strictly proved ; but he maintains 

 that it can be applied only to such expressions as are suscep- 

 tible of no more than one development. By means of this 

 modification he arrives at the same conclusions which are 

 stated by Disjota in this Journal for last month (p. 84-). But no- 

 thing can be more clearly proved than that no function can pos- 

 sibly have more than one development ; which sets aside the sug- 

 gestion of the Professor ; who must, therefore, be ranked with 

 those that admit the unrestricted theory of Laplace. Mr. 

 Bowditch, in his excellent Translation, limits the general equa- 

 tion of Laplace, which applies to an attraction proportional to 

 the n\\\ power of the distance, by excluding all negative values 

 o{ n from —2 to — oo ; and, by so doing, he has brought this 

 curious but slippery speculation one step nearer the grasp of 

 the human mind. Confining his attention to the law of at- 

 traction that prevails in nature, he attempts to prove the ac- 

 curacy of Laplace's method in all its generality, drawing his 

 arguments chiefly from geometrical considerations. It will be 

 sufficient to remark here, that the nature of the function ex- 

 pressing the height of the molecule does not depend upon 

 any integral taken between very small limits; but, as Laplace 

 has clearly stated*, upon this, that the differentials shall in- 

 variably continue to be infinitel}' small as the molecule ap- 

 proaches the contact of the two surfaces. It will not be ne- 

 cessary to examine all the demonstrations enumerated ; for 

 they all turn upon the sense, more or less extensive, in which 

 one equation is to be understood. It will be sufficient to dis- 

 cuss the investigation of M. Poisson, vhich is adopted by 

 M. de Pontecoulant in his Traite du Systeme du Monde. 



The question is fairly stated by Disjota in tlie last Number 



of this Journal. The value of X', which is the limit of the 



integral X when a = 1, is shown to be a series, determinate 



in its form, all the terms of which satisfy an equation in partial 



• Mec. Cel., toni. v. pp. 25 & 2G. 



