84 On the Rev. J. H. Pi-att's Demonstration of 



the lifter never can exceed that of the electro-magnet, and 

 consequently never can induce a higher magnetic state in it 

 than what lias already been done bv the voltaic hehx. 



XVII. Remarks on the Rev. J. H. Pratt's Demonstration of a 

 Propositioti in the Mecanique Celeste. Bj/ A Coruespon- 



D£NT. 



To the Editors of the Philosophical Maga:;ine and Journal. 



Gentlemen, 

 A T page ^T^ of your Journal for last month (June, vol. viii.) 

 "^^ an old controversy is revived. Whatever may have been 

 intended by bringing forward this subject at the present time, 

 it is certain that no new light is thrown upon it. 



In order to separate what is clear and undeniably proved 

 from the point in which lies the difficulty, it will be necessary 

 to set out from M. Poisson's statement oi the problem, viz. 



^ _ 1 p^ r^^ ( l-«-^ )/( fl',rt,')sin9 ' rf9'rf4;' 



(1 — 2aj9-f a^)2 

 p = cos 9 cos 6' -|- sin 9 sin 6' cos {•^—']>') ; 



the arcs fland -^i are the initial values of the variable arcs 9' 

 and ^l/', which increase from 9 and vp to O + tt and \{/ + 2t; and 

 a is a constant which is here supposed less than 1, but may 

 a})proach to it indefinitely. These things premised, if any 

 value be assigned to a, the symbol X stands for the definite 

 integral taken between the extreme limits of the variable 

 arcs. 



1st. Let the expression under the sign of integration be 

 expanded in a series proceeding by the powers of «, then 



X = -^ nr^ '/' (9', i,') sin fl' d 6' d'^' 



{1 +3«P, + 5«2P2 ... (2/ + l)P,... }. 

 For the sake of simplifying put 



Y, = '1^ /"" y^^ " p./(9', vj/') sin ^<d^'d ot' ; 



then 



X = Yo + « Y, + «^ Y2 a' Y, ... . 



All the coefficients, P, , satisfy a certain equation in partial 



