120 Dr. Brinkley’s Investigation of the general Term 
f 2 3 
A ecu + A/3 =1 h+Ay f?+ AS j- 3 h 3 + See. 
B^h+B[2-i i h 2 +B'yl 3 h 3 + See. 
n ^ 23 
4 -C a 4 ^ 2 +C^|A 3 + &c. 
3 
Da 4; AM- &c. 
&c. &c. 
Hence 
Aa — 1 
AjQ = o 
Ay B/2 -{- Ca = o ( 3 ) 
A<^ -j- By -}- C/3 -J- Da = o 
&c. &c. Sec. 
In order to obtain the values of a, / 3 , y, &c. let 2 repre- 
sent a function of h which expanded gives 
a -f ( 2 h -j- yh z 4 - + &c. or % = a -f jQA + y/i 2 + 2 h 3 See. 
Then by equations (3)' we have, when h = o 
?n 
m-i 
m— z 
m-i 
m 
( 4 ) --- ^2: vz + vz -{ vz -{- vz = o, m denoting any 
number. For taking m successively 1, 2, 3, &c. and dividing 
by h, h % , k 3 , Sec. we obtain the equations (3). Now the equa- 
tion (4) is the mth fluxion of 
(5) - vz = constant, divided by 1.2.3 • - m - When li—o 
vz = Aa — 1, hence const. = 1, therefore vz = 1, or % = — = 
( 4 ir=('+^+^+ &c -'-”* 
* It is hoped, that this investigation of the value of z is stated with sufficient 
clearness. It is a very simple example of a method of very extensive application 
with respect to analytical functions. The theorems for finding fluxions per saltum , 
communicated by me to the Royal Irish Academy, in November, 1798, and published 
