186 MR. GRAVES ON THE INACCURACY OF SOME LOGARITHMIC FORMULAE. 



Note K. To exemplify the agreement with which the positions we have established lead by dif- 

 ferent processes to the same conclusion, it may be mentioned that the same general formula for the 

 Neperian logarithms of 1 would be obtained from [23], on supposing y=l,c = 0,a = e and c = -v/ — 1 , 

 or, more concisely, from [26], on supposing K= = 1. 



If, however, in [23] we had selected other values for c and c, consistently with the convergence of 

 the numerator and denominator ; e. g. if c were supposed =3 2 j ff and c = 2 t ir + v' — 1 , upon 

 making all the necessary substitutions ; formula [23] would produce 



2 ! ir — 2 « ir 



2iir— 2lir — V — l 

 Now though this formula has precisely the same values as 



IV 



P=y yet their arrangement is 



different. In general, therefore, [23], from its liability to alter the arrangement of its values by the 

 alterations imparted to c and c, cannot be resorted to for the definitive computation of the orders and 

 ranks of logarithms. It was from the necessity of establishing a standard (whose only requisite is 

 that, when once determined, it should not be varied,) from which to commence such computation, that, 

 in Appendix, § 5, 1 fixed arbitrarily (the consideration of superior simplicity abstracted) on that value of 



cos 



— 1 



R 



a which I denote by cos 



-1 



R 



which would satisfy the equation 



jy/R? + S» 



f cos 



-1 



although any other defined value cos 



_ tt + V-iS 



-1 



R 



Vii' + s*' 



VK» + S> ^&» + S« 



would have answered the same purpose. 



When R is negative and S = 0, according as we decide to consider positive or negative. 



,-1 



R 



i will = either + « or — w ; in every other case the value of cos 



-1 



R 



55 £5 Will be 



^R' + S' 



definitively fixed by [29]. 



If 1 a designate the O"' Neperian logarithm of a of the 0*'' order, [32] may be expressed as follows ; 



^—I2iv + In 



which may be compared with (3) and (4). 



