294 



1 J 



F(M) = F[^(a;)] + xF'[<^(a;)]f(*) +— — xV[^(a;)]^'(^)+&c 

 wherein 



From this result the author deduces the theorems of 

 Laplace and Lagrange for the expansion of implicit functions; 

 and he shews that, through the arbitrary character of ^ (x), 

 they are particular cases of a class of theorems, infinite in 

 number. Applied to the solution of the equation /"(m) = x, 

 by making f(m) = u, this method gives that root which will 

 be represented by <j>(v), in which v is the least root of the 



equation 



f[^(v)]=x.. 



The President read the following letter from Mr. Cooper: — 



''Markree Castle, July 3, 1846. 

 " My dear Sir, — 1 now beg to transmit to you, for the 

 favourable consideration of the Academy, the observations we 

 have been able to make on comets at this Observatory, 

 during the first six months of this year. They are preceded 

 by the places of some stars with which we compared the 

 comets, and which we were forced to determine, as they were 

 not included in any catalogue we possess. The results of 

 observations, made by Mr. Graham principally, for polar 

 point on circle, are also added. The dates without places, 

 signify that we have the observation, but not yet the stars of 

 comparison. 



" Believe me, my dear Sir, 



" Your's very sincerely, 



" Edward Cooper. 

 " The Rev. H. Lloyd, D. D. 

 Sfc. Sj-c, Sfc" 



