Ii8 Dr. Brinkley’s Investigation of the general Term 
~nh 
A" u=e x 
— [n—\)h / „ , \ — In — z)h — A 
■ ne x —J— ( — -\e x — +i=(e x — i)” 
substituting as above-mentioned. 
n k + 1 n _j_ t ti -\- 2 n i 2 
Theorem II. A” u= ~ If 4 - a — — h 4-b— h 
X* 1 .M+I 1 .«4- 2 
M-fm 
M — 
■n4- m 
v * 
/z -f- & c - where M is the coeff. of h in the ex- 
pansion of (e b —i) n or of h m in the expansion of (~j~)”°rof 
( 1 +T£ + IX* + 
~h 
Dem . By Theor.l. a n u={e x — l )* making the necessary 
change after expansion. This change does not affect the nu- 
merical coefficients, which are evidently the same as those of 
( e h — i )*, and because ( e h — i y=h n ( t + ~ + 77 : +> &c.)” 
the theorem is manifest. 
n . 4 ni n + m 11 + m 
Theorem III. The coefficient of h =A" o 
•n-\- m 
x 
Dem. The successive values of n are 
2 2 
u, u+ — h -f- di /z a + &c., u + 4- eA-f- 2 * /z 2 -j-&c., zz-f- 4- $h 
X X mm . X X ■ ■■■ X 
2 
+ 4 3’ A*+ &c. and therefore (the whole ;zth differences 
being made up of the zzth difference of the parts) we have 
K 771 n + m n + m 
A” zz= 4- hA ” 0 + -1 /z 2 A" o 2 -f 
n + m 
h 
A” o 
-f- &c. 
where it is to be observed that whilst n+m is less than n 
A" o ” + m = 0 , and A” o"=i, as is well known, and easily ap- 
pears from Theorem II. 
