124 On some Elemeiitary Applications o/* Abel's Theorem. 



theorem relative to the sum of three arcs of an equilateral hy- 

 perbola. 



The coefficient of — in the development of 

 x^ lo J ^1-fr-^ I 



= coefficient of — in the development of 

 X ' 



"1 - M[,-K^.,|,_^,,^.}| 



— JTj 0*5 X^, 



Hence (seep. 118, line 5,) 

 f, / — ==r + £a / — + g^ / — =^ — = constant 



T" ^j X(^ Xg 



/vi±z dx^-^ ^^^" + 2 r^'jf, , 



^ x" X J -v/l+o;* 



and, since = — = V c. 



' X X X ^ 



"^ *^ ^ 



111 Sx, ajo Xq 

 = —^ " -:;r - 'o - + 2^^ x^Xg + constant 



= — jL'i ^2 X3 + constant, 



which is Mr. Talbot's theorem. 



In the numerical examples offered in this paper, I have 

 only carried the approximation as far as could be done con- 

 veniently by the common tables of logarithms, that is, to six 

 places of decimals. 



If L be the logarithm of the number N, L \-x the loga- 

 rithm of the number N+i/, 



