[ * 34 ] 
XVI. Two general propositions in the method of differences. By 
Thomas Knight, Esq. Communicated by Taylor Combe, 
Esq. Sec. R. S. 
Read February 27, 1817, 
i- Though so many ingenious writers have demonstrated, 
and, in various respects, extended the celebrated formulas 
of La Grange, for A n <p (.r), % n <p {sc) no one appears to have 
entertained the idea, that these, and the more general cases, 
in which the quantities under the functional sign have their 
differences variable, might be included in one simple form. 
Mr. Prony* is, I believe, the only mathematician who has 
given a form of any regularity to A” (p{x ), when the diffe- 
rence of x is variable ; but he does not seem to have been 
aware of the capability of the method he was employing ; 
and instead of embracing, as he might have done, all cases 
in one simple expression, he has proposed a formula which 
has neither any particular elegance in itself, nor any appa- 
rent relation to that which, in the simpler case, had been 
given by La Grange. 
I suppose the truth of the differential equations 
A>=<s--® +^=^ 4 , — (”— z > $ +> 
^ 1 T 1.2 — 2 1.2.3 K— 3 1 ’ V ) 
«(«+!) 
<P 
.2.3 «— 3 
. n(n+i) (»+ 2 ) 
^ i ^ + j) T~ lt2 r (« + 2) ■ 1.2.3 
where <p is any variable function whatever. 
* Lacroix “ Calc, des Diff.” p. 25. 
~(« + 3 ) 
+ v .(2 
