126 Dr. Brinkley’s Investigation of the general Term 
m -f 1 term is the coefficient of h m in the expansion of 
( 1 +T^+7T- 3 + &c.)-' 1 =( 1+ i*A +1 _*fr + &c.) 
Letrf (,) =_i” ,+I 
^ Z) = J (e) ( (i!)”’ w h ere / + V — z,pt -f qvz=m-\- 2, 
unity being excluded from the va- 
lues of p, q, &c. 
(i*) v [f_) W , where t-\-v-\-w=z^,pt-\-qv~\- 
4-rze==w~f-3, unity &c. 
Then the coefficient of k m =-nd ^ n (n-\-i ) d ^ — n [n -f- 1 ) 
(«-f 2) J. . n (H-J-i ) • • + w — 1 ) <i ( ' m \ 
This may easily be deduced from the multinomial Theorem, 
or more readily from the theorems for finding fluxions per sal- 
turn in the seventh volume of the Transactions of the Royal 
Irish Academy. 
By Lemma II. 
</ (, W o’"+' 
7(2) a 5 M + 2 .. W2 + I 
d K =A 0 —A 1 0 
,(3) . 3 "2 + 3 nz + 2 , 1 * m+x 
d^=A 0 — A 2 0 -1 A 0 T 
■ 1.2 
A m 2m in — i 2m — i , i . m — z 2m — 2 
:A 0 —A 0 4-— A 0 
__ ' 1.2 
&c. 
Whence the coefficient of lf ~ 
n 
— A 1 o m * + «(«+ l) 
A* o ’ 71 ~ + w (22+ 1 ) (22 + 2) 
«(« + 1) 
«(«+0 (» + 2) 
m(?2+x)..(m+3) 
k(«+i)(h + 2) 
h(«+ 1 . .(« + 3) 
«(«+l)- (22 + 4) 
I 2 
1.2 
1.2 
(m terms) 
(m — i terms) 
(in — 2 terms) 
