294 
e(u)= Fp @)] +3" [6@)] 9 @) +5 T@)1¥@)+8e 
x = o— f[p(2)]. 
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 $(a), 
they are particular cases of a class of theorems, infinite in 
number. Applied to the solution of the equation f(w) = 2, 
by making F(z) = uw, this method gives that root which will 
be represented by ¢(v), in which v is the least root of the 
equation 
wherein 
S1¢(&)] = 2. 
The President read the following letter from Mr. Cooper:— 
** Markree Castle, July 3, 1846. 
«¢ My pear Sir,—I 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, 
** EpwarpD Cooper. 
“« The Rev. H. Lloyd, D, D. 
Fenge cer 
