324 



celeration. The expression wliich he thus obtains for the first two 

 terms of this acceleraticn, is 



According to Plana, the c:rresponcjn2: expression is 



It will be obsei-ved that the coeiiicient of the second term has been 

 completely altered in conseouence of the introduction of the new 

 terms. 



The numerical eftect of this alrfr:.::::: i? to diminish by l"'66 the 

 coefficient of the square of the time ::i :he expression for the secular 

 acceleration; the time being, as usual, expressed in centuries. 



It v.-ill, of course, be necessary to cany the approximation much 

 further in or;":: such a r:..:u:- ::' this : : efficient as may be 



employed wim conh^ieiice in the Cai:v.i::i::: :: imcient eclipses. 



In conclusion., the author states. : he existence of the new 

 terrr.s in rh:- expression of the moon s c::rdinates occurred to him 

 ■ - ti n:- nn::: v i::n i:e was engaged in thinking over a new me- 

 thc;. :: ::::::::- In:::.r theory, th:urh he did. not then perceive 

 then- imp .:m.n: " :mi::r :n the secui:.:' epnr.tion. His attention was 

 t'-rs: : ire ::■ tnis sn: ect while enm:." :::ring to supply an omis- 

 sic't in :i:e rn£::y :: :i:: :!::.: pi-en iry Pontecouiant in his 

 ' I---:ri: Anaiytique.'" In this vmu.-.:ie "nrk, the author, follow- 

 ing the ex?.mple originally set :p ni: J, Lubbock in his ti-acts on 

 the lunai- theory, obtains dnectiy the expressions for the moon's 

 coordinates in terms of the time, which ?.re found in Plana's theory 

 by means of the reversion of series. With respect to the secular 

 acceleration of the mean motion, however, Pontecouiant unfor- 

 tune.teiy [.icpts Plana's result without examination. On perform- 

 ing the ceicularion recuisite to complete this part of the theory, 

 the author was surpri;::! m hnd that the second term of the expres- 

 sion for the secular acceleration thus obtained, not only differed 

 tcte.llp in magnitude from the corresptndins: term given by Plana, 

 but nn.s e-.-en :i :. centi-ary si:rn. Kis -re-n-ns researches, however, 

 immetiiet i " led him t: suspect whet m:.s tit : eriidn of this discord- 

 ance, :.::i mhen : :th ::::;^:5 were ccrrectc-tl e y taking into account 

 the new terms whtse existence he had aire :.dy reeegnized, he had 

 the satisraction of hnding a perfect agreement between the results. 



3. " On a Theory of the conjugate relations of two rational inte- 

 gral functions, comprising an application to the Theory of Sturm's 

 Functions, and that of the ^rreatest Airehraical Common r^Ieasure."' 

 By J. J. Sylvester, Esq., MA., F.R.^., Barrister at Law. Received 

 June 16, iS53. 



The memoir consists of four sections. In the nrst secrion, the 

 theoiy of the residues obtained by applying the process of the common 

 measure to two algebraical functions is discussed. It is shown that 



