536> Mr, Gompertz on the nature of the function 
If log. of is less than 3,00, then it will be necessary to 
calculate A by common methods, as the tables do not 
go ower. And generally it will be then sufficient, omitting 
a only to calculate the value of- — x (a—i) ; but from this, if 
more accuracy be required, subtract the number whose 
common logarithm is (1,6378 + X (r)^). 
^ (tit) given, and X (^) be required, proceed thus, 
I— a 
10 
^ ( a'-i I t>eing = ,89144 for example. In Table I, the next 
value of 
^ ( a — I ) which X (a *°) corresponding is 1,79 
Difference ,00256 belonging to T.79 gives . . ,005 
Difference ,00020 . . . ditto .... ,0004 
* * ^ ^ ( TT7 / “ >^9‘44 then we have X (a =r 1,7954 
If X [a^) is less than 3, proceed thus : put the given value 
of X = we have the common logarithm of 
a:=z^p X X(i +q~^) + a small correction if great accu- 
racy be required ; which correction is nearly equal to 
p X the number whose common log. is {* -6378— ^?—p+T: + 
These methods and tables only apply immediately to 
X when ^ is a proper fraction ; but if a be greater 
than unity, put it equal to b , then will 6 be a proper fraction ; 
1 ^,,^ 1 — at. I b-i P + 
71 ^ - — = = 
quently =^+i. X(/>)+ X I have likewise 
had Table IV. calculated, which is a general table, for the com- 
mon log. of 1^:^), corresponding to a given value of Xa, 
