in the method of differences. 237 
A*(o+or=(«„+n' ii r— *y—’ 
the successive values of 0 being u ,u ,u , .u 
123 n 
those of o' being w ,w ,w , w 
0 1 2 3 n 
, . , _ i , n , n(n — i) n(n—i)(n — 2) . 
observing therefore that l — + — — + 
=A»(o+o')% and putting 2 (■———) for <p [x ,y) our equa- 
tion takes this form 
a^,>)=s(^^)sa”(o+0’+s(^ ( ^)eia*(»+oT+ 
J_ s (jtv&2L)M A*(o+</) ! + TJ . S (JWiL'Wa «(o+o') 3 + . 
2 \j, z - m dy m S ' , 
What has been done with respect to a function of two vari- 
ables, the analyst will immediately see how to extend to a 
function containing any number ; we may therefore without 
entering into any farther particulars, give the following 
General Rule. 
Let the successive values 
of 0 be u , u , u , u 
123 n 
of o' be w , w , zv w 
123 n 
of 0" be v , v , v v * 
123 n 
... Sic ,then zvill 
A n <p(x,y, z, See.) = A” e o+o'+o"+&c. 
provided that, after the expansion, we multiply every where a term 
of the form A x«V r x w n _ r x v c n _ r x &c. by 
* Supposing z n —%~v n , &c. &c. 
