i /— 
— Ss eh llUC<C 
—— 4 _—_ 
287 
¥* (2) =S( Ff (@) + 2} 
¥%(a) = fF (@) +3} 
¥" (@) = fF (a) + 2} 
Hence, amongst other consequences, it is evident that any 
functional equation of the form 
An" (%) + Ana, ¥"" (a) +....4+ A(z) + Ax =0, 
in which 4,, An, &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 y (x) since 
o() =F f(a) - 1} 
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 
d d d 
ti ee 
e ern ag dz 
in which LZ, M, N, are functions of z, y, and z; and which 
has the effect of changing «, y, and z respectively into certain 
functions of x, y, and z, whose form depends upon that of LZ, 
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- 
_ euracy 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 
