Mr. Rumker on the Formula for clearing the Lunar Distance.169 



if this is less, but subtracted from it if it is greater, than 90, 

 gives the true distance. 



Demonstration. 



D = apparent distance L S. 

 k apparent sun's altitude, 

 H = apparent moon's altitude. 



Then is, with the omission of 

 the third correction, which we 

 shall explain hereafter, the true 

 distance Is = LS Lm+ Sm 

 = LS-L/cosL + SscosS. 



-o . T sin h sin H cos D sin h 



But cos L = 



cos H sin D " cos H 2 sin J D cos J D 



sin H sin h sec 2 \ D-sin H (1 -tang* j- D) _ 



cos H tang D 2 cos H tang \ D 



sin ft sin H -f (sin h + sin H) tang * \ D _ 

 2 cos H . tang \ D 



s ; n '*- sln H + tang 9 i D tang H( S ! n *~ S !" "cotangSD+tangSD) 

 sm h + sin H \sinA-fsinH / 



cotang H . 2 sin H tang \ D sin h -\- sin H (sin h sin H) 



siu A -j- sin H sin /i -j- sin H 



tang H ftaDC | D -f ^^nS.. cotang J D^ 



* sin A -|-sin H / 



sin /t sin H 



cotang I D . tang \ D 



tang h (*-H) cotang J. D 



1 - tang i D - 



. _ tang ^ (A-H) 



and making tang A = cotang | D tangf(A+H) 



we have cos L = tang H . tang ( JD A) accordingly as 7^ ^ H. 



and also cos S = tang li tang (4 D + A). Q. E. D. 



In case that A > ^ D, the sign of the cosine of either L or 

 S, and consequently that of the corresponding correction, will 

 be changed. It may easily be proved that A is the part of 

 the apparent distance intercepted between its middle and a 

 perpendicular from the zenith upon it. It remains now to 

 explain the third correction, which is nearly applicable to all 

 approximate methods : 



We have hitherto supposed s 1 = s m, which is incorrect. 

 N.S. Vol. 9. No. 51. Mar. 1831. Z Describe 



