INACCURACY OF SOME LOGARITHMIC FORMULAE. 
181 
Hence some value of sin 1 0 or in must be to be found among 
A [2 i 7r 4 - cos - 1 + B1 V' x 2 
Hence 
1 */ x 1 must = some quantity 
x is either positive or negative. 
When x is positive, cos ~ ~^=~ 2 = chs “ 1 or 0 
/ . ^ _ | X 
i 7T — A 2 i TT + COS — 7=; 
\ V x 
') 
[30] 
B 
• \ __ 1 x x 
When x is negative, cos — = = cos 
\ „ - 1 
1 or ± t. 
Hence and from [30] it follows, that, for y as well as x to be real, x must 
= one of those quantities whose tabular Neperian logarithms are 
i — 2 i A 
B 
or one of the negatives of those quantities whose tabular Neperian logarithms are 
i — (2z + l) A 
" g ” 
Hence x must = one of the quantities 
[31] 
a formula, which, as appears by [11] and [27], comprises all the quantities that 
respectively fulfil the conditions above stated. 
Corollary. On retracing our steps under the guidance of formula [31], it 
would not be difficult to prove, among others, the following theorems, viz. 
1st. When B = 0, for x to be negative and y real, A must be a rational 
fraction with, in its lowest terms, an odd denominator. 
2nd. When B = 0, and A is a rational fraction, which, in its lowest terms, 
7fl 
= — , the number of real values of y that can correspond with a real x will 
be one or two, according as n is odd or even. 
3rd. In general, when A is irrational, y can have only one real value con- 
sistently with the simultaneous reality of an x. 
4th. When B is not = 0 and A is rational, y , in every case when it has one 
real value corresponding to a real x, has an infinite number. 
