470 PHILOSOPHICAL TRANSACTIONS. [ANNO 1799. 



mean log. value of gb, we shall obtain a very accurate correction of the assumed 

 latitude. But if there be more observations on one side of the meridian than on 

 the other, when all the pairs have been united, and the areas resulting from them 

 found, we may combine the supernumerary observations on either side with any of 

 those which are made on the opposite side. The fittest however for this purpose, 

 is the observation which is made at the least distance from the meridian. I should 

 hope further, the practical astronomer will think it a circumstance of some moment, 

 that the principal part of the work consists in finding the time, an operation which 

 he is obliged so frequently to perform. Any of the 3 methods which are usually 

 adopted on this occasion might easily be applied to the tables which have been de- 

 scribed; but I shall venture to recommend a different rule, which I conceive to be 

 better adapted to our purpose than any of the others, and to which the directions 

 before given had a particular reference. 



Let a be the sine of the altitude; y the cosine of the hour-angle; d the sine, 

 $ the cosine, and r the tangent of declination; / the sine, a the cosine, and 5 the 



r . 1 1 , • 1 rrM. a — dl a dl ars .a . x 



tangent of the latitude. Then y = -^- = ^ - g = ^ - rs = (^ - 1) . rs, 

 when radius is unity, but = (-£- — ra 2 ) . — , when radius is m = ^x into the 



v dl ' m l m* 



square of the tangent of the arc whose secant is */ -jr. Hence we deduce the fol- 

 lowing rule for determining the log. cosine of the angle at the pole. From the log. 

 sine of the altitude increased by 3 times the log. radius, subtract the sum of the 

 log. sines of the latit. and declination; take half of the remainder, and, consi- 

 dering it as the log. secant of an arc, find the log. tangent corresponding; multiply 

 this by 2, and add the log. tangents of the latit. and declin. and reject thrice the 

 log. radius; the sum will be the log. cosine of the angle required. But, when the 

 declin. and latit. are of different denominations, it is evident that our expression 



G7H, T*S f*Q *• 



becomes (wi 2 -f- -tt) . — s , which is equal to — X into the square of the secant of 



the arc whose tangent is */ -r-. In this case therefore, having found the log. value 



of -jr, and divided it by 2, we must consider the quotient as the log. tangent of an 

 arc, whose log. secant being taken, we are to proceed as in the former case. 



The advantages of this rule are obvious. We obtain the angle in terms of the 

 log. cosine; and consequently, when we have calculated the 2d time with the new 

 latit. we have only to subtract one result from the other, and we immediately deter- 

 mine the area corresponding to the difference of the times. Besides, in the 2d 

 computation, fewer of the elements are changed by this rule, than by any of those 

 which are usually employed; and this is a consideration of much importance. But, 

 if we are disposed to adopt the following method of ascertaining the incremental 

 area gc, this advantage will be found still greater. Let us resume the expression 

 y = ~^~ , and we shall have V = j X ~ m - ax + — *_ ^ ^^g t fo e succeeding 



