344 .JJHILOSOFHICAL TRANSACTIONS. [anHO I766. 



XXIX. Theory of the Parallaxes of Jltiiude for tfie Sphere. By Mr. F. 



Mallet, Professor and j^stronomer at Upsal. From the French, p. '244. 



Let p be = the moon's horizontal parallax, or I to sin. p, as the moon's dis- 

 tance to the radius of the terrestrial sphere, on which the spectator is supposed 

 to be placed. Let a be the distance of the moon from the zenith, and p the 

 parallax of altitude for the same distance. The astronomers usually compute the 

 value of p in the following manner: let sin. p = sin. p sin. a, and />' being found 

 by the tables of logarithmic sines, sin. p" = sin. p. sin. (a -|- p') is found in like 

 manner, p" being assumed for the true parallax, which is not accurate. 



2. In order to show this, Mr. M. has given another method of computing the 

 parallax of altitude as exactly as may be, by means of the common tables, in the 

 following manner: since sin./) = sin. p. sin. (a -f p'), we have sin. p = sin. p sin. 

 a cos. P + sin. p COS. a sin. p, or sin./) (1 — sin. p cos. a) = sin. p sin. a cos. 

 6; hence tang, b = '■ . This formula seems a little difficult to be 



J^ ^ o /^ 1 _ sin. p COS. A 



wrought in numbers, but it is as easy as the one above: for, supposing sin. b^ = 

 sin. p COS. A, the tables will give the angle b, and tang, p = -i^:-^'-" -i, the com- 

 putation of which can give no trouble. Hence it appears, that the calculus for 

 finding the true parallax is not more difficult than that which gives the said pa- 

 rallax with an error, the value of which is unknown; for it is evident that the 

 above computation for finding p'' is only an approximation, and that to make it 

 accurate, it would be necessary to carry it still on by finding sin./)'" = sin. p sin. 

 ^ _j. //"), and afterwards sin. p"" =■ p sin. (a -j- />'") &c. 



3. Mr. M. therefore thinks himself in the right to prefer this method to that 

 hitherto used by astronomers. To confirm his opinion, he made a trial, by put- 

 ting P = 59' and A := 30°, and found p — //' = 0'.43, in which the error of the 

 usual computation amounts to near half a second; he therefore gives the prefer- 

 ence to the geometrical calculus. 



4. Before he quits the formula tang, p = ■ — ■ , he observes, that the 



' o i- 1— sin. p COS. A ' 



computation of p may be executed by other methods to the same exactness. If 



we take cos. 2 c = sin. p cos. A, we shall have tang. 6 = -7- '°"r » and the 



"^ ' 2 (sin. c)* 



computation of this new formula is extremely easy. 

 Ci. The formula tang. /) = 



> ""' ^ ^ ~— — , gives besides, sm. p = - , . — =; — -~ ; make sin. p =: 



1— sin. p COS. A ° ^ VI + sin. p' — 2 sin. POOS. A 



1 cos. D, D being a given angle, of which we may have tables ready made, and 



sin. p sin. a sin. p sin. a 



we shall have sm. 6 = ■ — ^''P^r : = "/ ." .""^"^ "^ — I — rT"*""- 



^ VI + 2sin. p (cos. D — cos.a) Vi + 4 sin p .sin, Ja + ^d 



(4-A — in); since cos. n — cos. a = 2 sin. (^a -|- -^d) sin. (-J-a — ^^d). This 

 being found without any logarithmic computation, we shall find tang, e^ = 4 sin. 



