10 SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 66 



fore that the corrections belong- only to a, h' (the initial point), or to 

 au and /, which is brought about by a rotation arourtd the apparent 

 terminal point, a", 8". This rotation must be so made that 

 [/.(Aa'«.cos=8' + A8'*)] 



is a minimum for the necessary corrections cos 8/ • Aa/ and ASV, tak- 

 ing into consideration their respective weights Pi. 



To this end, again just as in the case of the determination of the 

 geographic co-ordinates of the terminal point, we proceed best by 

 graphic methods. We adopt an approximate position for the point 

 {a, d) which we can here designate by {ao, do) and coiupute or 

 graphically measure its normal distance A( from each apparent orbital 

 path uki, Ji. 



If we regard A as positive, when {ao, do) lies within the northern 

 polar region of aid, Ji, and if we wish to find this distance as correctly 

 computed, we use the following formula : 



sin Ai= —co&Ji sin Jo + sin /» cos do sin(ao— o^). (12) 



Now since the A<'s represent discrepancies, the corrections 

 Aao cos do, ^do 



must be so determined fliat those shall be zero. 

 Putting 



sin /( cos(oo — afc() =cos Pi, 

 cos Ji sec do = sin Pi', 

 Aa,. cos do = X, At/,. -'- V 

 sin Ai = Ai, 

 the condition for this is 



o = A(— A"COsPi+y sin P('. (14) 



In order to properly express the equations of error, we must still 

 consider the fact that A, or the change of location that the provisional 

 radiant must experience in order to fall in with the observed orbital 

 path, is not an obseri'ed quantity since the corrected place to be 

 obtained by the rotation is (o', 8') . Let ^ be the change that must be 

 produced by the rotation, then we have 



sin C ^ C _. sin/ . . 



sin A A sin/" ^ ^ ^^ 



where / denotes the length of the arc from (a, 8') to (o", 8"), /' that 

 from (ao,do) to (a", 8"). 



(13) 



sin / 

 sm /' 

 tion as the weight. If we omit this, then the short orbital paths have 



Therefore the factor '^"^ ., = V^ is to be appfied to the whole equa- 



