INACCURACY OP SOME LOGARITHMIC FORMULA:. 181 



Hence some value of sin ~ * or i it must be to be found among 



A (2 IT + cos " ' -^) +81-/^ 



Hence 



eT-A(2iV + c6s-^^,) j-g^-j 



\ s/ x' must = some quantity g 



X is either positive or negative. 



When X is positive, C()s ~ -j=% = cbs ~ 1 or 0. 



1 *r 1 



When X is negative, c6s ~ — = = c6s — 1 or ± ir. 



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 



£ — 2 / A 



B 



or one of the negatives of those quantities whose tabular Neperian logarithms are 



i-{2i± 1)A 

 . g ^ 



Hence x must = one of the quantities 



f{(.-^-i^).} [31] 



a formula, which, as appears by [ll] and [27] , comprises all the quantities that 

 I'espectively 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 

 fi*action with, in its lowest terms, an odd denominator. 



2nd. When B = 0, and A is a rational fraction, which, in its lowest terms, 

 = ~, 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 = and A is rational, y, in every case when it has one 

 real value corresponding to a real x, has an infinite number. 



