of Trigonometrical Quantities from one another. 153 



3. In applying the formulae, let it first be supposed that a 

 log. cosine is given, to find the corresponding sine : Then, putting 

 # + A# for the angle corresponding to the given cosine, the 

 next greater in the table will be cos x ; this, therefore, as well as 

 A log cos x ( log cos (x + A*) log cos x), cot x, cosec x, and 

 sin x, will be given. Let us for a moment put 



y = A log sin x, p = co\.x cosec x cos (x + A#) ( A log cos #), 



then formula (B') will stand thus, 



here p is a known quantity, and we must find such a value of y 

 as satisfies the equation. 



Now, y is always small, because it is the logarithm of a quan- 

 tity differing but little from an unit ; therefore, as a first ap- 

 proximation, 



Log y = log p, and y = p nearly ; 



and hence 



Log y = \ogpp nearly. 



If the value of y, found from this last equation, be not suffi- 

 ciently exact, it is at least a nearer approximation than the first 

 assumption y =p ; therefore, denoting it by y', we shall have still 

 more correctly, 



Thus, it appears, that to approximate to the value of y in the 

 equation 



~Logy \ogp- p, 



we have only to form a series of quantities y', y", y", &c. from the 

 function p, such, that 



y=p, iogy=iogp y, iogy = iog^ y, & c . 

 and these will be successive approximations to the value ofy. 



Next, let us suppose that a log. sine is given to find the cor- 

 responding cosine ; let x -f- A# be the angle to the given sine ; 

 the next less in the table will be log sin x, and their difference, 

 (viz. A log sin x) will be given, as well as tan x, sec x, and cos x. 



VOL. x- P. i. u 



