242 Mr. Rumker's new Method of reducing 



for their central altitudes, whereas it is the refraction of those 

 points of their limbs whereof the distance is actually measured 

 that should be used in the calculation. As introduction, it 

 may not be useless to remind, that the usual methods of 

 clearing the distance, which suppose the altitudes known by 

 observation, can be classed in direct and approximatory ones. 

 For as all the former methods are derived from the equation 

 existing between the sides of the two triangles formed by the 

 apparent zenith distance and apparent distance, and the true 

 zenith distance and true distance, and only differ by slight 

 variations in the manner of finding thence the true distance, 

 or side of the latter triangle opposite to the angle at the zenith 

 which is common to both triangles, thus all the latter methods 

 agree in the former or apparent triangle, the angles at the sun 

 and moon, and the sides adjacent thereto in two right-angled 

 triangles ; having for hypothenuses the corrections of the sun's 

 and moon's altitudes, these sides are the corresponding correc- 

 tions of the distance by a first approximation. 



Let D designate the apparent distance of centres, S and M 

 the above angles at the sun and moon, g — n the differences 

 between parallax and refraction or corrections of altitudes, 

 then is the true distance of centres = D + cosine S.(§ — tt) 

 + cosine M (§' — »') + « {g-^f + jS^ - Jf + +. 



The moon's parallax being greater than her refraction, the 

 second correction becomes negative; and when one of the 

 angles is obtuse, its cosine takes the opposite sign. The fun- 

 damental formula of all approximatory methods is, therefore, 



.. _. /sine h — sin H cos D\ , N 



true distance = D + I -. — fj—. — fv — J (e— *) 



v cosin H sin D ' V5 ' 



/sin H — sin h cos D N , ; 

 + ( cosin a sin D )(*—> + +» 

 where g and it refer to that altitude h or H which stands first 

 and by itself in the parenthesis. Lions obtained by executing 

 the division, 



D + (sin k secant H cosecant D — tang H cotang D) (g — n) 

 + (sin H secant h cosecant D— tang h cotang D) (g — v), 



_ ... /sin h — sin H cosin D\ , L 



By calling I -,— u -.- n — 1 (v-*) 



" & v cosin H sin D ' V5 * 



/ , sin h— sin H cos D\ , v 



(1-1+ • — ft— i — n — J (§-*) 



* cosin H sin D ' V5 



/ 1-2 cosin £(D + H + £)sin£(D+H--#h • 



■ V cosin H. sin D t V"*' 9 



