Sixty-five Stars near 61 Gygni. 117 



28. Let the angles and arcs have the significance attached to 

 them in the figure ; where P is the Pole ; PA and PB hour- 

 circles through 61 1 and 61 2 respectively at 1873.546, and M'M 

 and 2V 7 .AT the paths of proper motion. Then, as the arc AB joining 

 61 x and 61 2 is the mean of all the measures of distance, it maybe 

 assumed as the true distance at the mean date of observation. 

 In like manner PAB is assumed as the true position-angle at the 

 same date. The problem before us is to (1) determine the distance 

 MN at the end of the interval r years ; (2) determine the angle 

 PMN at the end 'of the same interval. In both these cases evi- 

 dently we regard the onty cause of motion of the stars to be that 

 known as proper motion, i.e., uniform motion on the arc oi a 

 great circle. 



Let all angles count from the hour circle positively towards the 

 east, i.e., counter-clockwise in the figure ; or else from the arc 

 AB, but always likewise positive when counter-clockwise, as 

 shown by the direction of the arrows in the figure. 



29. Let us first suppose that 61 2 remains motionless while 61 1 

 advances from A to M during the time r; then the change in dis- 

 tance is given by the formula : 



Aff = ( r Po)c<*(*o— #0) 



+ Oo) 3 1 — ~ 2 sin 2 K — Jo) cos K — , To ) J 



+ (-Po)* J — 3 [1- sin* (tt — xo) — sin 2 (tt — Xo ) cos 2 (tt — * )] j . 



When the proper constants have been substituted in this we have: 

 A<7 = [9.657224](Tp )4-[8.3H234„](T/) )a- r -[6.68iO5„](7/> )3-|-[3.63io Jl ](Tp )«. 

 Thus is obtained the auxiliary distance BM agreeing of course 

 with the formula of paragraph 11, as far as terms in r 2 . 



30. Now during the same interval of time r, suppose 61 1 to re- 

 main still and 61 2 to be in motion. Then the change of distance 

 is given by the formula : 



A/ff o=( 7 /V)cosCt ' — z) 



+ ^ 2 {- 2 K^) sin2 ^-^} 



+ (-Po / ) 4 { 2((7o ^ A( . o)3 [isin*(AV- g )-siD 2 (,Y o ^-0) 008* Uo 7 -*)] } 



