Mr. Bums's Reply to Mr. Riddle and Mr. Henderson. 345 



of which the three sides are given to find sPZ. Again, with 

 the polar distance augmented or diminished by the declination 

 in the intenal, we have another triangle SPZ, whose three 

 sides are given to find ZPS. The sum or difference of these 

 angles ought to equal the interval in degrees, &c.; and if it do 

 not equal it, make PZ vary a little, and we should soon find 

 a side PZ to represent the observations. Thus by a little 

 trouble the apparent time may be determined, as well as the 

 latitude. What Mr. Riddle means, when he calls the inter- 

 val itself an assumption, in my solution, I own I am unable to 

 divine, and must therefore leave to himself to clear up. Does 

 he mean that the problem can be solved independently of the 

 consideration of the interval? If se, it is plain that he has 

 indeed " misapprehended " it. 



It may be objected that the above method is indirect : but 

 even so, it will be frequently found shorter than most of the 

 methods that have been given : and the advantage of its de- 

 termining the horary angles is peculiarly its own, — angles 

 which these gentlemen seemed to think could not be ascer- 

 tained. 



Another method, which is direct, and in which the altitudes 

 and the interval only are assumed, may be had, by means of 

 the following formulae, which it will be easy to deduce, from 

 tlie properties of spherical triangles. 



Let SP = a 

 sV = b 



S s = c 

 Zs = d 



zs = ^ 



S P 5 = 771^ 



ZP = y 



P S 5 = A 



Z.S5 = B 



A-t-B = C 



sin.^ ^ = sin. a . sin. i .cos.* -^ 



— = s-m. (-^— + P) . i-i"- ( — f/ 



sin. A = 



sm. -— = 



e — c). sin. i(c -^- d — e) 



= sm. a . sm. e . cos. 



sin. |j/= sin. (^-j- + ^^sin.^-^^" ~ *). 



in which ?, and 5, are what are denominated subsidiary angles. 

 This method is direct and rigorous, and by means of it 1 as- 

 certained that the latitude (bund by Dr. Brinkle\''s method 

 was nearly correct; — but I also found that the intei val given 

 in the Doctor's examjjle (2) was incoirect; and hence arose 

 the great difference in the latitudes determined by his method 

 and mine. And when the data are wrong or incongruous, it 



Vol.66. No. 33 1. 



• Equal the intenal 

 -W. 1825. 



Xx 



