287 



^0*)=/-M/(*) + 2} 



$"(x)=f-i{f(x)+n} 



Hence, amongst other consequences, it is evident that any 

 functional equation of the form 



An \p n (») + An-x, i^"" 1 (x) + ■ ■ ■ • + A xp (as) + A x = 0, 

 in which ^„, ^4 n -i» &c., are constants, may be reduced at once 

 to a linear equation in finite differences with constant coeffi- 

 cients. 



We might also invert the function \p (x) since 



f»-/- T {/to-i) 



Dr. Graves stated that, in a continuation of the present pa- 

 per he would lay before the Academy the results which he had 

 obtained in discussing the symbol 



g dx dy dz 



in which L, M, N, are functions of x, y, and z ; and which 

 has the effect of changing x, y, and z respectively into certain 

 functions of x, y, and z, whose form depends upon that of L, 

 M, and N. One example of this kind has been already com- 

 municated to the Academy in a paper read by Dr. Graves on 

 the 9th of June, 1851. 



The Chair having been taken by the Rev. Dr. Lloyd, 

 The President communicated the results of four years' ex- 

 perience, at his own observatory, of the effects produced by 

 the vicinity of a railroad. 



" Amid the ever increasing requirements of improved ac- 

 curacy which the progress of science is pressing on astronomi- 

 cal observers, it becomes important to avoid every possibility 

 of error ; to remove every cause that may, in the slightest de- 

 gree, add to the difficulties that inevitably oppose our advance 



