4 REPORT — 1859. 



6 to 6+1, 



6+1 an — ab — n — b—l ,s 



n — b' ab — b ' 



b to 6+2, 



(6+1) (6+2) {^^7b-(n-b-\)\\an-ab-(n-b-2)} ^ 



(n-6)(»-6-l)' (a&-6)(a6-6-l) 



These results are capable of remarkable interpretations, -which will be best illus- 

 trated by a numerical example, which by the aid of logarithms may be worked out 

 with great ease *. 



On Calculating Lunars. By Colonel Shortrede. 



Besides the trigonometrically rigorous methods of reducing lunars, there has been 

 during the last ninety years a multitude of approximate methods more or less exact, 

 and no lack of subsidiary Tables for facilitating the solution. 



The method here proposed is short and simple, and requires no subsidiary Tables 

 beyond those of refraction and parallax, and the common Tables of logarithms to 

 five places; and the result is always correct to within a quarter of a second. 



The corrections in altitude into the cosines of the adjacent angles give the prin- 

 cipal corrections on the apparent distance. We cannot from the given altitudes and 

 distance get the cosines directly, but we find them by means of the versines. The 

 rio-ht-angled triangles formed by perpendiculars from the true places to the apparent 

 distance being calculated as plane triangles, the greatest possible error on the moon's 

 correction cannot exceed 0" - 4. 



The smaller segment of the true distance is found by means of the logs used in 

 finding the principal corrections, using as a first approximation the reduced or cur- 

 tate distance instead of the greater segment. The cotangent of the greater segment 

 to three places requires to be found ; for the cot of the smaller segment we may 

 use the complement of its log sine. 



The squares of the perpendiculars are found by taking the product of the sum 

 and difference of the corrections in altitude and in distance. 



Logs to five places are required for the principal correction of the moon ; for the 

 other parts of the work, logs to three places will ordinarily suffice. 



For the error on M/x as above found. 



tan M/x= tan Mm . cos m, 



M^+ M^ + &c. = cosM(Mm+ —■ + &c). 



Mu-Mm cos M= ^ cos M- -^ 

 r 3 3 



= ME 3 C osM-M!2lcos'M 

 3 3 



= M^!cosMsin 2 M. 

 3 



For secondary corrections. 



cos .cosp=cos/t=cos (6+d) = cos 6 cos d— sin 6 sin d, 



. ■, . i i , 7 v t « • V-\-d ■ v — a 



sin o sin «= cos 6 (cos a— cosp) = cos6 . 2 sini . sin > 



2 Jt 



sind=cot6 (vers p— verse?) =2 sinHpcot6— cot6 versd, 



0,,_-iinJl» 2cotJ _ &c< 



2 r 



* The above paper is published at length in the Philosophical Magazine for November 

 1859, and in the Assurance Magazine for January 1860. 





