go Mr. Hellins's Second Appendix to the improved 



than the common logarithm of ' , even with the use of 



Taylor's excellent tables, is too obvious to need a description ; 

 and therefore it follows, that a computation by these new for- 

 mula will be easier and shorter than by those in the first 

 Appendix. 



6. But there are still some expedients by which the compu- 

 tations of A, B, &c. may be further facilitated and abridged. 



It is pretty evident, to any one who contemplates the coeffi- 

 cients of the logarithmic terms in the first three theorems, that 



the terms a + \&cc -f -—j- ac A , in the first theorem, being once 



found, the logarithmic terms of the second and third theorems 

 may most easily be derived from them ; in consequence of 

 which, the greater part of the time of writing down the loga- 

 rithms of -/-, -fi, \, and T 9 T , of twice writing down the 

 logarithms of ace and ac 4 , and of searching in the tables for 



the numbers corresponding to -^-acc, and 3 s 21 ac*. in the 



i o 4.12 ' 4.12.32 



second theorem, and for those which correspond to -slcc, and 

 -~r at -4 , in the third theorem, is saved. These are the expedients 

 which I am next to explain. 



7. The three terms a, J-acr, and -^ac 4 , which are found in 



the first theorem, are evidently to the three terms j|a, -^-acc, 



and z ac 1 ", which are found in the second theorem, in the 



4.12.32 ' 



ratio of 1 to -f , 1 to |-, and 1 to % respectively ; or as 1 to 1 — £, 

 1 to 1 — ^, and j to 1 — -§- ; by which mixed numbers, the 

 logarithmic terms in the second theorem may more easily be 

 derived from those in the first theorem, than by the fractions, 

 as will appear further on. 



