186 MR. GRAVES ON THE INACCURACY OF SOME LOGARITHMIC FORMULA*). 
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 = — 1, 
or, more concisely, from [26], on supposing K 2 = 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 = 2 itr and c = 2 l it + V — l, upon 
making all the necessary substitutions ; formula [23] would produce 
2 lit — 2 i ir 
2 i it — 2 lit — V'—l 
i it 
yet their arrangement is 
Now though this formula has precisely the same values as - , 
2 iit — V — 1 
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 which I denote by cos — 1 - / s r - although any other defined value cos — 1 ■ ■ 
yy/R- + S 8 J y/R 8 + S 8 o j .y/R 8 + 
which would satisfy the equation 
c „ _i R _ R + S 
t COS + S 8 ~ yy/R 8 + S 8 
would have answered the same purpose. 
When R is negative and S = 0, according as we decide to consider 0 positive or negative, 
R 
y/k 8 + S 3 
definitively fixed by [29]. 
will = either -f it or — it ; in every other case the value of cos 
v — 1 
R 
y/R- + S 3 
will be 
If 1 a designate the 0 th Neperian logarithm of a of the 0 th order, [32] may be expressed as follows : 
y/— 12 IT + II/ 
y/ — 1 2 i TT + 1 a 
which may be compared with (3) and (4). 
