226 
Mr. Knight on the construction 
(»-3))+ +( L"-»— (W 1 )LM*-0+ 
3 ? z) — K( ”- 1 1 ) ^ ,, ~ 5) L''- («-3) L . /•••« -f 
-j- L.y'--(« + I ) ; any immediate term, as them-}- i fi , will be 
of the form L (m — l )L"--(”— 0 -|- " • X [™~~ 3) L' , '( w -- :2 > ___ 
OT (m— i) x(m— 5) , * 
• 2 -3 
We easily see that 
r 
(« + i)x(n— 2) 
L.5 * 
:L-< 
(x + n)x(x + n— z) 
X(x+n— 4) 
(n + 1 )«(« — 1 ) x ( n — 6) 
1. 2.3.4 
1 
X &c. J 
(w+l)«X(«— 4 ) 
1 - (x-fn— i) M x(tf-l-w— 3) 1,2,3 X &c. J 
It is scarcely necessary to say that L . «"•••*, L . s" « are to 
be expanded by the common series for L . viz. 
* These forms are analogous to an expression in the method of differences, which, 
though not noticed by Stirling and other writers on interpolation, may be useful 
on many occasions, as the coefficients are small and few in number. Borda’s ex- 
pression for logarithms is a particular case of it. 
.3) n(n—i)x(n—s\ 
-«„_ 2 + — * 
njn—i) (k— 2 )xp?— 7) , 
1.2.3.4 
+ * + A”m + A”« —1 . If we make u n =L . x, we have, by taking n (in the co- 
efficients) successively 1, 2, 3,4, &c. 
L • xxxL ( x — 2)+ series, 
L . x~L (x — i) + L(x— 2) — L(x— 3)4- series, 
L . x—z | L(x — 1) — L(x — 3) | -f L(x — 4) + series, (Borda’s if we change x into x + 2). 
L .*= 3 | L(x— i) + L(x— 4) | —2 | L(x— ■ 2) + L(x— 3) j — L(x— 5)+ series, 
L • | L(x — 1) — L(x— 5) | — 5 | L(x— 2) — L(x — 4) j +L(x — 6) + series, 
L . xz=5 | L(r— i) + L(x— 3> + L(x— 4) + L(j— 6) L(x— 2) + L(x— 5 L(x— 7( + series, 
L . xzz 6 | L(x — i)— L(x — 7) j-i 4 [ L(x— 2— L(r— 3) + L(x— 5)— L(x— 6) j 4- L(x— 8) 4- series, 
&c. & c . 
which may be useful, when taken without the series, as formulas of verification. 
