Mr. Knight on the construction 
228 
— 1) 
1) 
-p whence L . a l} =L . « a 
>-0 
by means of which equation, L(jt-|-w + i) becomes 
L(x + K+ 1 ) =L(x + «) H-L.» + -2-L.«'+"-^L. £e "+ 
1-2.3 * 
+L.«'+^L a "+^L., , " + 
or L (x + n -f- i ) = L(# -f «) + L . a-f- ? -^p-L . . «'+ 
(” + 1 ' "(”“i) L . 4 - the same form as before, which is there- 
fore general. 
9. By means of equations (r) the logarithm of a number 
is found from the logarithm of that which precedes it, and 
the process will consist of a number of additions, equal to the 
number of the values of a. we make use of. 
Thus, if L . u ,u be so small, as not to affect the last figure, 
to which we intend to carry the logarithms, it may be neg- 
lected and we must make 
L . *+ L. x‘ =r 
L.*+2L.*'+ L cc"=r +L .«'+ L.«" =r' 
L . x-\~sL . *'+sL • «"=r' +L*' +2L . a" =r" 
L . *-HL . «'+6L . «"=r"+L . «'+3L . *" =r'" 
Here the quantities r', r", r‘" , See. are formed by two addi- 
tions each, one more gives the logarithms; for, by substituting 
in (<?), 
L . x =L(.r — i)-j-L.a 
L(x-}- 1) =L . x — f- La - j- 
— L . jr-J-r 
