128 Dr. Brinkley’s Investigation of the general Term 
Example . To find the first term of the n — 2 order of dif- 
ferences of the series T 2 , {x-\*h) n , (a 1 -{-2 h) n , &c. 
By Theorem III. 
ti — 2 n , x 27 n — 2 n — 2 k — 2 , / x in — l 
A x = n ( n — 1J..3 xh A 0 — 1)**2 xh 
.71 — 2 71 — I , , X 7 _ M — 2 w 
A 0 — 1 ) . . in" A 0 
and, By this Corollary 
A 0 ~[n — 2 ) A 1 o 2 =— p- 
A ? ‘“ 2 0" — 2) (?z — 4) A'o 3 -f («— -2) (« — 3) A*o*= 
Hence 
. (»— 2 )( 3 « — ?) 
I-2-3-4 
A *”=1.2 
-£l hA 
1.2 • 2 1 X 2 3.4 / 
Scholium. 
The case of the above theorem, when n—i, has, on account 
of its importance, been a particular object of investigation 
among mathematicians. Although the formula in the first 
corollary is, as to its formation and law of progression, very 
simple, yet one more simple may be readily obtained by the 
joint application of a transformation given by M. Laplace in 
his method, and of the first lemma above given which for- 
mula and its investigation are here subjoined. This and the 
above general formule ( n being any number), as well as the 
formula of Laplace ( n being 1), do not enable us to compute 
the successive coefficients so readily as from the equations of 
relation. But this circumstance, it is imagined, will not render 
what has been here done less worthy of the notice of mathe- 
maticians. Their researches for a general term in the case of 
