124, Dr. Brinkley’s Investigation of the general Term 
n (i) 
1 = a> ' 
n (1) , (2) 
2 = 2<r -j- a> 
n (1) , (2) . (1) 
3 = 3 a + 3 a + a 
n (1) . m(m—i) (2) (m— 1) , (m) 
in = m a> H — a r m a> '4- a> 
These equations correspond to the equations between the 
terms of a series of quantities, and the first terms of their 
respective orders of differences, i. e. o", i”, 2", &c. correspond 
to the series, and a^\ f z \ &c. to the respective orders of dif- 
ferences. 
Hence we conclude, that a^ — A m o\ and therefore that 
f( if)' (if) &c. = 4" o\ 
Lemma II. Every thing being, as in the preceding lemma, 
except that unity is excluded from the values of p, q, r, &c. 
f \ At I Av I Aw o A m n m— i n— i , i . m— z n— 2 
J T l? l"l &c. = A 0 — A 0 A 0 
— A™ -3 0’ 1 - 3 (to m terms.) 
Demonstration. Let b'' m \ &c, represent 
1.2 . . m f[ lP ) t ( 1? ) U &c. ; 1.2 . . [m — 1 ) J ( ( 1 ?Y &c. ; 
&c. &c. respectively : these latter quantities being defined, as 
in the preceding lemma, except that unity is excluded from 
among the values of p, q, &c. 
Then it is easy to see, if a^ m \ , &c. denote as in the 
preceding lemma, 
that a [m] = b {m] + + ’^=TL 
1.2 
-f- &c. 
