XOL. LXXXIX.] PHILOSOPHICAL TRANSACTIONS. 47 1 



. x 1 1 «AT«« <fc«x -«&+«#* dAx a-d* x 



value of a) = 7 X —, = fep - ^r 3ST" x 3T = 



(— — y) . i = (taking a positive instead of negative, as it ought to be when / is 



A' 



r\..*,. „„j ~- — +« y _ /, r \ > 



positive), (-y )•— J an( l consequently - = (l ) . -, when radius is unity; 



but = (m* — — ) . -At;, when radius is m. Now - may be considered as the in- 

 crement of the hyperbolic log. of y, and therefore, with its proper modulus, may 

 represent the area which is the object of our investigation. We may suppose the 

 other side of the equation to be the square of the cosine of the arc whose sine 

 i Sv / — x into the increment of the hyperbolic log. of a 1 , divided by the square 

 of the radius; and if, instead of taking this log. with the hyperbolic, we take it 

 with Briggs's modulus, we must then consider - as the increment of the log. of y, 



according to the same system. But -? being equal to — (when i is only 1'), it will 



vary as s; and therefore, if its value be determined according to Briggs's system, 

 when s is equal to radius, and be denominated v, its value in any other case will be 



expressed by — . 



Hence, to obtain the log. of the area gc, the quantity with which we are imme- 

 diately concerned, we must find the log. value of — , and divide it by 2; we must 

 then take out the log. cosine of the arc whose log. sine is equal to the quotient; 

 and, having multiplied it by 2, we must add the product to the constant log. of v 

 (3.1015), and the log. tangent of the supposed latitude, rejecting thrice the log. 

 radius. But if \/ — be greater than radius, which must necessarily be the 

 case when the azimuth is greater than a right angle, we must then consider 



rrn" « _. ,, c t u„ i. 4. „r *i ...i ^ • yrm 3 



VI m 2 as the square of the tangent of the arc whose secant is \/—, observing 



* in other respects the directions before given. The quantities r and s are both em- 

 ployed in the first computation, from the result of which we also obtain y; and 

 consequently this operation will not be attended with much trouble. 



The above instructions, it is manifest, are given on the supposition of r and s 

 having the same sign; but if the declin. and latit. should not be of a similar deno- 



fini Vic 



mination, then will our expression become (m 2 -| ) . — , and we must consider 



m 1 -| as the square of the secant whose corresponding tangent is y^ — . With 



this exception, the process will be the same as when the tangents r and s are both 

 affirmative. 



The preceding formula naturally suggests to us another method of finding the 

 log. area gc; and as some perhaps may think this more eligible than either of the 

 former, I shall take the liberty of explaining it. When the latit. is given, the 

 area gc, it is obvious, must invariably preserve the same magnitude at all distances 



