Dr. Brinkley’s Investigation of the general Term , &c. 115 
4-z> 
&c. ; and provided that in the expansion of 
— n 
(e* — 1 ) , fl. 
?z_i n — 1 . 
ux , &c. be substituted for 
— 71 I 
&c. and -L, 4-, &c. be substituted for (A-j , j4-j ,&c. 
. These theorems, which M. Lagrange had not demonstrated 
except by induction, have since been accurately investigated 
in different ways by M. Laplace,* and also by M. Arbogast'|\ 
The expanded formula for S n u, or, more accurately speak- 
ing, the natural series for S”w is of the form 
fl.” ux tt -\- ^rr fl.” 'ux 1 1 
. . \ fl. nX +1/M -f 7T 4 - 
&C. 
The coefficients a, /3, y, &c. are readily obtained by equa- 
tions of relation, which were first given by Lagrange. But 
to complete the solution it is obviously necessary to obtain 
the law of progression, and be able to ascertain any coef- 
ficient independent of the preceding ones. This has not 
hitherto been done, as far as I know, except in the case of 
n—\. M. Laplace has given a very ingenious investigation 
for that case. J This has been copied with just encomiums by 
M. Lacroix, § who does not mention that any one had accom- 
plished it for any other value of n. In this Paper the general 
term is given for any value of «, the law of which is remark- 
ably simple. A particular formula, remarkably simple as to 
its law, is also given for the case of n~ 1 , from which that of 
Laplace may be deduced. 
* Mem. Acad. Scien. 1777 and 1779. 
f Arbogast du Calcid des Derivations, Art. 395, See. 
t Mem. Acad. Scien. 1777. 
§ Lacroix Traite des Differences et des Series, Art. 918. 
Q* 
