of Logarithmic Tables. 
223 
For, if it be denied, let this represent a single result, to 
pass on to the next, we change x into x— 1, in the logarithm 
of the fraction, and add a new logarithm (L) to restore the 
equality : the equation will thus become 
n[n— 1) ") J w(h — 1) (n— 2) ~j 
>L( x— 1 ) h(x— 2) >L(x— 3)— ....+(L) 
4-1 I — - -1 
L . x=’^x-i)-^^L(x-2)+ ( iUH. , '- ,) ' 
and, by transposition, we find 
C (« + »)» 
x= T | 
1.2.3 
■L(x — 3)— ....-{-( L), 
(L)=L< ■ l>;( - r_21 ' 2 x(i - 4) 
{« + i)»(w 1) (« — 2 ) 
1-2-3 4 x &c. 
«+i (« + !)«(«— 1) | 
t (*— 1) 1 X(x— 3) 1,2-3 x&c. ^ 
so that the whole expression is of the same form as before, 
which is therefore proved to be general. 
6 . Cor. 1. If in the values of x. in the last article, we put 
for x, in the second jc— j— 1 , in the third x-\-q, in the fourth 
and so on ; and moreover represent the last fractions 
arising after such substitution by «, al , u " , a!" , &c., we get 
the following set of equations 
X = (X — l) x a 
X -j- 1 = X x a Hoc 
0 ) 
* + 3 = 0 +®)* 
&c 
which , from the manner of their formation, are subject to this law, 
that the m th fraction (provided it is not the last) in the value 
of x-\ -n, is equal to x-\-n — 1 divided by the product of the 
first m— 1 fractions in the expression of x-\-n — 1. 
Gg 
X+l 
/ // 
X 
X 
« X « 
■*’ + 2 
*(x + z) 
JT+I 
X 
(*+>)* 
MDCCCXVIIT. 
