THE DISTANCES OF A COMET FROM THE EARTH. 99 



coordinates, the following method may be applied to refer them all 

 to the system to which the original values of a, 6, &c. (2 a) belong. 



Let P be the pole of this system, and let A and D, A and 

 D, be the angles corresponding to a and 6, by which are refer- 

 red to P the two points B and B\ determining the position of (ii) 

 the plane in (10) from which the transformations are to be made. 



PjB = 90° — A P5' = 90° — £>', BPb = a — A, BPB = 

 A' — A, BPb=A — G, ftp = 0o= the per- 

 pendicular from h upon ^5'; b P = 90° — cr, 

 Bb^:=^(o, b B' B =^£2, &c. : we are to find an 

 expression for sin. 6^, for any position of the 

 point b, in terms of A, A, D, D', a and et, 

 observing that sin.- d^ may have any coefficient 

 which remains constant in all positions of b, because this will 

 disappear when sin. 6^ is introduced into (4). 



We have sin. 0Q = sin. a sin. Si, and by multiplying both sides (la) 

 of this equation by sin co", it becomes 



sin. ^o sin. 6)" = sin. w' sin. to" sin. SI. 

 sin.^ 6o sin." eo" — sin.'' w' sin.^ to" sin.' J1={1 — cos.'' to') {1 — cos.'' to") ( 1 — cos.- Jl). 



Substituting in the last member cos. ji — ''°^' 3""^!. T' "' , (12) becomes 



sin.- 60 sin.'' co" r= 1 -}- 2 cos. w cos. to' cos. w" — cos." to — cos.'' to' — cos ."to". (12o) 

 cos. to ^sin. D sin. cr-f-cos. D cos. to cos. (a — .4), cos. to'r=sin. D' sin. s7-|-cos. D' 

 COS. nr cos. {A' — a), cos. to" = sin. D sin. D' -j- cos. D cos. D' cos. (A' — A). 



These values of cos. a, cos. a, and cos. a" being substituted in 



(12 a), it becomes, after reduction (see Memoirs of the Berlin 



Academy, for 1783, p. 308 et seq.), 



sin. ^o sin. to"= [sin. {A' — o) tan. D — sin. {A' — A) tan. nr + sin. (a — .4) tan. D'] 

 cos. D cos. D' COS. ru ; 



by means of which equation (4) becomes, by substituting suc- 

 cessively the proper values of sr and <r, 



