of a Series in the inverse Method of finite Differences. 117 
- — — is denoted by - 1” 
1.2.3 - n J _ 
is denoted by - - -= - (i n ) m 
(i.2.$..n) m l-z. 3. .m J \ f 
According to which notation, 
e — 1 1 — -j- 4 & c - = 1 -f* 1 J r^. 2 +J_ 3 + 
e x = 1 -J- x -f- — -f- “I” = 1 -{- x 4 af 4 
If the quantity (if)' (if)” &c. or (rrm^FTTZTT^ 
has various values arising from different values of p , q, t, v, 
&c. then the sum of them all is denoted b y/( if)' ( 1.)’ &c. 
Theorem I. Let u be a function of x and xf-h, 
4 h 
x-\~nh successive values of x. Then A "u — [e x —1)" if after 
the expansion of this latter quantity 
2 3 
U U 
&c. be substituted 
for ( 4 ) , ( 4 ) 3 , See. respectively. 
Demonstration. Let the successive values of u be repre» 
sentedby u, u, u, . . . u.u. Then bv Taylor’s theorem. 
123 («— 0 » 
3 
4 nh 
u—u -f- — nh 4 4 2 n*ti- 4 ^n 3 h 3 + Sec. — u 4 e * i,sub- 
stituting as above-mentioned. In like manner, 
4 (»— 
«— 1 
u=u -j-e 
•1, substituting &c. 
— (n—2)b 
u—u-\-e x — 1 &c. &c. 
(«— 2) 
4 h 
u—u 4 ~e x —1 Sec. Sec. 
1 1 
u—u -\- 1 — 1. 
Hence u — 1 being common to each term, we have by the 
differential theorem. 
