714 Table of Proportional Logarithms. [No. 117. 



Exa?nple. — The Table Diff. for 10' being 461, required the propor- 

 tional part for 4' 10". 



Opposite 4'. 10" (column' ") is the logarithm, 6198 

 Opposite 461 column N. is 8855 



The sum is, 5053 



The next less log. in the Table is 5051, corresponding to 192, and 

 for the difference 2, the table of proportional parts gives *!, hence the 

 whole proportional part is 192*1. 

 Required the log. sin. and tan. of 22° 27' 37".3, using Button's Tables. 



For 37-3 Proportional Log. is, ..7936 | .. 7936 



For 3U56the Tab. Diff. for sin P.L. 7069 



N 1899-6 50U5 

 Log. sin 22° 27'= 9-581 9236 



Lo2-. sin 22° 27' 373" =95821 135-6 



Tab. Diff. for tan. 3579 P.L. 7756 



N 2225 5692 

 Ltang. 616 1514 



9-616 3739 



It very often happens, that the correction for 2d difference is omitted, 

 though it may be sufficiently large to affect the result. To make this 

 correction as little troublesome as may be, I have prefixed a set of de- 

 cimal factors, which multiplied by the second difference will give the 

 correction to be applied, with a sign opposite to that of the 2d dif- 

 ference. For example in Button's Tables, the log. sin. of 22° 27' has a 

 2d difference of 3. The coefficient for 2d difference at 37-3 is -119, this 

 multiplied by 3 is '357 or -4 ; which added to the result above found, 

 gives 9-5821136 as the log. sin. of 22° 27' 37".3. 



The Table here given has no indices. The want of them may be 

 supplied by the following Rule. When the fractional part of 10' for 

 which proportion is required is between 10' and 1' the result is greater 

 than j^Q of the Tabular Difference, when between 1' and 6" the result 



is between j^^ and fo^ of Tab. Diff. 



When the Tab. Diff. is for 1', then between 1' and 6" the result 

 is greater than f^ of the Tab. Diff. and similarly in other cases. 



P.S — In using this Table to find the Logistic Log. for 1 hour, the frac- 

 tional interval is to be reduced to decimal of minute, and found in 

 column N. ; the Logistic Log. is the arithmetical complement of the 

 rithm in the Table. And similarly, if the whole term be 3 hours, 



